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DIFFERENTIAL  AND   INTEGRAL 
CALCULUS 

WITH   EXAMPLES   AND   APPLICATIONS 

BY 

GEORGE   A.    OSBORNE,   S.B. 

WALKER    PROFESSOR    OF    MATHEMATICS    IN    THE    MASSACHUSETTS 
INSTITUTE    OF    TECHNOLOGY 


REVISED   EDITION 

OF  THE  ^'. 

/ERSITY 

Of  y 

BOSTON,   U.S.A. 
D.    C.    HEATH   &   CO.,   PUBLISHERS 

1907 


OA 


7 


Copyright,  1891  and  1!K)6, 
By  GEORGE  A.   OSBORNE 


PREFACE 

Ik  the  original  work,  the  author  endeavored  to  prepare  a  text- 
book on  the  Calculus,  based  on  the  method  of  limits,  that  should  be 
within  the  capacity  of  students  of  average  matheuiatical  ability,  and 
yet  contain  all  that  is  essential  to  a  working  knowledge  of  the  subject. 

In  the  revision  of  the  book  the  same  object  has  been  kept  in  view. 
Most  of  the  text  has  been  rewritten,  th'i  demonstrations  liave  been 
carefully  revised,  and  for  the  most  part  new  examples  have  been 
substituted  for  tlie  old.  There  has  been  some  rearrangement  of  sub- 
jects in  a  more  natural  order. 

In  the  Differential  Calculus,  illustrations  of  the  "derivative" 
have  been  introduced  in  Chapter  II.,  and  applications  of  differentia- 
tion will  be  found  also  among  the  examples  in  the  chapter  immedi-. 
ately  following. 

Chapter  VII.,  on  Series^.is  entirely  new.  In  the  Integral  Calculus, 
immediately  after  the  integration  of  standard  forms.  Chapter  XXI. 
has  been  added,  containing  simple  applications  of  integration. 

In  both  the  Differential  and  Integral  Calculus,  examples  illustrat- 
ing applications  to  Mechanics  and  Physics  will  be  found,  especially 
in  Chapter  X.  of  the  Differential  Calculus,  on  Maxima  and  Minima, 
and  in  Chapter  XXXII.  of  the  Integral  Calculus.  The  latter  chap- 
ter has  been  prepared  by  my  colleague.  Assistant  Professor  N.  R. 
George,  Jr. 

The  author  also  acknowledges  his  special  obligation  to  his  col- 
leagues,  Professor   H.  W.   Tyler  and  Professor  F.   S.   Woods,    for 
important  suggestions  and  criticisms. 
January  1,  1907. 

iii 


CONTENTS 
DIFFERENTIAL   CALCULUS 

CHAPTER   I 

Functions 

ARTS.  PAGES 

1.    Variables  and  Constants 1 

2-7,  0.    Definition  and  Classification  of  Functions 1-5 

8.    Notation  of  Functions.     Examples 6-7 

CHAPTER  II 
Limit.     Increment.     Derivative 

10.  Definition  of  Limit 8 

11.  Notation  of  Limit 8 

12.  Special  Limits  (arcs  and  chords,  the  base  e)  ...               8-10 
13-15.  Increment.     Derivative.     Expression  for  Derivative    .         .         .11, 12 

16.    Illustration  of  Derivative.   -Examples 13-15 

17-21.    Three  Meanings  of  Derivative 16-21 

22.    Continuous  Functions.     Discontinuous  Functions.     Examples    .     22-25 

CHAPTER   III 

Differentiation 

24-32.  Algebraic  Functions.     Examples  .... 

33-38.  Logarithmic  and  Exponential  Functions.     Examples 

39-46.  Trigonometric  Functions.     Examples  . 

47-55.  laverse  Trigonometric  Functions.     Examples 

56.  Relations  between  Certain  Derivatives.     Examples 

CHAPTER   IV 

Successive  Differentiation 

57,  58.    Definition  and  Notation <'•! 

59.    The  »th  Derivative.   .  Examples 63-<55 

^  60.   Leibnitz's  Theorem.     Examples 65-67 


26-39 

39-45 

45-51 

51-57 

57-00 

CONTENTS 


CHAPTER  V 
Differentials.     Infinitesimals 


61-63.    Definitions  of  Differential 

64.    Formulae  for  Differentials.     Examples  . 
>j  65.   Infinitesimals 


68-70 
71-73 
73.74 


CHAPTER   VI 
Implicit  Functions 
3.    Differentiation  of  Implicit  Functioiis.     Example 

CHAPTER   VII 
Series.     Power  Series 


\ 


67, 68.    Convergent  and  Divergent  Series.     Positive  and   Negative 
Terms.     Absolute  and  Conditional  Convergence 

69-71.    Tests  for  Convergency.     Examples 

72, 73.    Power'Series.     Convergence  of  Power  Series.     Examples     . 


75-77 


78,79 
79-85 
85-87 


CHAPTER   VIII 
Expansion  of  Functions 

74-78.    Maclaurin's  Theorem.     Examples 88-93 

79.    Huyghens's  Approximate  Length  of  Arc        ....  93 

80,81.    Computation  by  Series,  by  Logarithms           ....  94-96 

82.    Computation  of  n- 96,  97 

V  83-87.    Taylor's  Tluorem.     Examples 97-100 

89.    Rolle's  Theorem 101 

90-93.    Mean  Value  Theorem 101-104 

94.    Remainder 105 


CIIAPTEIl    LX 

Ivni.TI.CMINATE    FoR.MS 

95.    Value  of  Fraction  as  Limit     . 

96,97.    Evaluati(m  of  -.     Examples 
0 

98-100.    Evaluation  of  g-,  O-cc,  x-x,  0",  1*,  x". 


Example 


106 
106-110 

110-11:1 


CONTENTS 


CHAPTER  X 

Maxima  and  Minima  of  Functions  of  Oxi;  Indkpendent 
Variable 

ARTS.  PAliKS 

101.    Definition  of  Maximum  and  Minimum  Values       .        .        .  114 

102-104.    Conditions  for  Maxima  and  Minima.     Examples.        .        .     114-111) 

105.  When  ^  =  3o.     F.xamples 110-121 

dx 

106.  Maxima  and  Minima  by  Taylor's  Theorem.     Problems         .     121-120 

CHAPTER  XI 
Partial  Dikikrentiation 

107.  Functions  of  Two  or  More  Independent  Variables          .         .  130,  1.^1 

108.  Partial  Differentiation.     Examples 131, 132 

109.  Geometrical  Illustration 133 

110.  Equation  of  Tangent  Planes.     Angle  with  Coordinate  Plane.s. 

Examples 133-186 

111,  112.    Partial  Derivatives  of  Higher  Orders.     Order  of  Differentia- 
tion.    Examples 1.36-139 

113.    Total  Derivative.     Total  Differential.     Examples  .         .  140-144 

114-116.    Differentiation   of  Implicit  Functions.     Taylor's  Theorem. 

Examples 144-147 


/ 


CHAPTER   XII 
Change  of  the  Variables  in  Derivatives 


117.  Change  Independent  Variable  x  to  y 148,  149 

118.  Change  Dependent  Variable 149 

119.  Change  Independent  Variable  j;  to  2-.     Examples           .         .  150-152 
120,  121.    Transformation  of  Partial  Derivatives  from  Rectangular  to 

Polar  Coordinates 152-154 

CHAPTER   XIII 

Maxima  and  Minima  op  Functions  of  Two  or  More 
Variables 
122,123.    Definition.     Conditions  for  Maxima  and  Minima  .         .     15'^,  156 

^  124.    Functions  of  Three  Independent  Variables    ....     150-161 

^  CHAPTER  XIV 

'        Curves  for  Reference 
125-127.    Cissoid.     Witch.     Folium  of  De.scartes         ....     162,163 


CONTENTS 


ARTS 


128-130.  Catenary.  Parabola  referred  to  Tangents.  Cubical  Pa- 
rabola.    Semicubical  Parabola 164,105 

131-134.    Epicycloid,    llypocycloid.    /^V"  +  A'^V' =1.     a^)j-'  =  a^x*-x'^     166, 1G7 

135-145.  Polar  Coordinates.  Circle.  Spiral  of  Archimedes.  Hyper- 
bolic Spiral.  Logarithmic  Spiral.  Parabola.  Cardioid. 
Equilateral  Hyperbola.     Lemniscate.     Four-leaved  Rose. 

r=asin3~ 167-172 


CHAPTER   XV 
^    Direction  of  Curves.     Tangents  and  Normals 

146.  Subtangent.     Subnormal.     Intercepts  of  Tangent  .         .  173 

147.  Angle  of  Intersection  of  Two  Curves.     Examples  .         .  174-176 

148.  Equations  of  Tangent  and  Normal.     Examples      .         .         .  176-179 

149-151.  Asymptotes.     Examples 179-182 

152, 153.  Direction  of  Curve.     Polar  Coordinates.     Polar  Subtangent 

and  Subnormal 182, 183 

154.   Angle  of  Intersection,  Polar  Coordinates.     Examples   .         .     183-186 
155, 156.    Derivative  of  an  Arc 186-188 

CHAPTER   XVI 

-    Direction  of  Curvature.     Points  of  Inflexion  -^ 

\ 

157.  Concave  Upwards  or  Downwards 189     \ 

158.  Point  of  Inflexion.     Examples 190-192 

CHAPTER   XVII. 

Curvature.     Radius  of  Curvature.     Evolute  and 
Involute 

157-161.    Curvature,  Uniform,  Variable 103, 194 

162-164.    Circle  of  Curvatuic.     Radius  of  Curvature,  Rectangular  Co- 
ordinates, Polar  Coordinates.     Examples  .         .         .     195-200 
166.    Cocirdinates  of  Centre  of  Curvature        .         .         ,         ,         .  200 

166, 167.    Evolute  and  Involute '      .         .         ,     201, 202 

168-170.    Properties  of  Involute  aud  Evolute.     Examples     .         .         .     202-205 

CHAPTER   XVIII 

Order  of  Contact.     Osculating  Circle 

171,172.    Order  of  Contact 206-208 

173.    Osculating  Curves 208, 209 


CONTENTS 


174.  Order  of  Contact  at  Exceptional  Points  ....  5»0!t 

175.  To  find  the  Coordinate  of  Centre,  and  Radius,  of  the  Oscu- 

lating Circle  at  Any  I'oint  of  the  Curve    ....     20'.»-211 

176.  Osculating  Circle  at  Maximum  or  Minimum  Points.      Ex- 

amples        211-21:3 


CHAPTER   XIX 
Envelopes 


177.  Series  of  Curves 

178, 179.  Definition  of  Envelope.     Envelope  is  Tangent 

180-182.  Equation  of  Envelope 

183.  Evolute  of  a  Curve  is  the  Envelope  of  its  Normals. 


214 

.  214,215 

.  215-217 

Examples    217-221 


INTECxKAL   C.\LCULUS 


CHAPTER   XX 
Integration  of  Stanuarh  Forms 

184, 185.   Definition  of  Integration.     Elementary  Principles 
186-190.   Fundamental  Integrals.     Derivation  of  Formulaj. 


.     223-225 
Examples    225-240 


CHAPTER   XXI 


Simple  Applications  of  Integration. 
Integration 


Constant  of 


191,192.   Derivative  of  Area.     Area  of  Curve.     Examples    .         .        .     241-244 
195.    Illustrations.     Examples 244-248 


CHAPTER   XXII 
Integration  of  Rational  Fractions 
194, 195.   Formulae  for  Integration  of  Rational  Functions.     Preliminary 
Operations 

196.  Partial  Fractions 

197.  Ca.se  I.     Examples -•'" 

198.  Case  II.     Examples -•^' 

199.  Case  III.     Examples -'»*' 

200.  Case  IV,     Examples 2(30- 


CONTENTS 


CHAPTER   XXIII 

Integration  of  Irrational  Functions 
arts.  pages 

202.   Integration  by  Rationalization 263 

208,  204.    Integrals  containing  {ax  +  h)i,  (ax  +  h)'.     Examples  .         .     263-206 

206,207.   Integrals  containing  y/ ±'x;- +  ax +  b.     Examples  .         .     266-268 

208.   Integrable  Cases 268 

CHAPTER    XXIV 

Trigonometric  Forms  readily  Integrable 

209-211.  Trigonometric  Function  and  its  Differential.     Examples       .  270-272 

212,213.  Integration  of  tan" X da;,  cot" a; da;,  sec" x(Zx,  cosec^xcZx  .  273,274 

214.  Integration  of  tan"' X  sec" a- dx,  cot™ X  cosec'xdx.     Examples  274-276 

215.  Integration  of  sin'"  X  cos"  XfZx  by  Multiple  Angles.    Examples  276-278 

CHAPTER   XXV 
Integration  by  Parts.     Reduction  Formula 

216.  Integration  by  Parts.     Examples 279-282 

217.  Integration  of  fi«*  sin  ?ix  dx,  e"^  cos  nx  fZx.     Examples     .         .  283,284 
218-222.    Reduction  Formulse  for  Binomial  Algebraic  Integrals.     Deri- 
vation of  Fonnuhe.     Examples 284-291 

223, 224.    Trigonometric  Reduction  Forniulse.     Examples     .         .         .     291-294 

CHAPTER   XXVI 

Integration  by  Substitution 
p 

226.  Integrals  of /(x2)xdx,  containing  (a +  6x2)?.     Examples       .     205,296 

227.  Integrals  containing  Va-  —  x''^,    Vx-  ±  a-  by  Trigonometric 

Substitution.     Examples 296-299 

228-232.    Integration  of  Trigonometric  Forms  by  Algebraic  Substitu- 
tion.    Examples 299-304 

233.  Miscellaneous  Substitutions.     Examples        ....  304, 305 

CHAPTER   XXVII 
Integration  as  a  Summation.     Definite  Integrals 

234.  Integral  the  Limit  of  a  Sum 306 

235-237.    Area  of  Curve.      Definite  Integral.     Evolution  of  Definite 

Integral 306-309 


CONTENTS 


238,239.   Definition  of  Definite  Integral.      Constant  of  Integration. 

Examples :]10-:>I4 

240-242.    Sign  of  Definite  Integral.     Infinite  Limits.     Infinite  Values 

oiJlx)        314-317 

243-245.   Change  of  Limits.     Definite  Integral  as  a  Sum      .        .  317-;iiy 


CHAPTER   XXVIII 

Application  of  Integration  to  Plane  Curves. 
Application  to  Certain  Volumes 

246,  247.    Areas  of  Curves,  Rectangular  Coordinates.     Examples 

248. 

249. 

250. 

251. 
252,  253. 


254. 


Areas  of  Curves,  Polar  Coordinates.     Examples    . 
Lengths  of  Curves,  Rectangular  Coordinates.     P^xamples 
Lengths  of  Curves,  Polar  Coordinates.     Examples 

Volumes  of  Revolution.     Examples 

Derivative  of  Area  of  Surface  of  Revolution.     Areas  of  Sur- 
faces of  Revolution.     Examples 

Volumes  by  Area  of  Section.     Examples       .... 


CHAPTER   XXIX 
Successive  IxTEGnATioN 
255-257.   Definite  Double  Integral.    Variable  Limits.    Triple  Integrals. 


Examples 


CHAPTER    XXX 
Applications  of  Double  Integration 
258-262.   Moment  of  Inertia.     Double  Integration,  Rectangular  Co- 
ordinates.     Variable  Limits.      Plane  Area  as  a  Double 

Integral.     Examples 346-.350 

263-265.    Double  Integration,  Polar  Coordinates.     Moment  of  Inertia. 

Variable  Limits.     Examples 3')0-;l.\:l 

266.    Volumes  and   Surfaces   of   Revolution,    Polar  Coiirdiiiates. 

Examples 35:{,364 


CHAPTER   XXXI 
Surface,  Volume,  and  Moment  of  Inertia  ok  Any  Solid 
267.    To  find  the  Area  of  Any  Surface,  whose  Equation  is  given 
between  Three  Rectangular  Coordinates,  x,  y,  z.     Ex- 
amples        


xii  CONTENTS 

ARTS.  PAGES 

268.  To  find  the  Volume  of  Any  Solid  bounded  by  a  Surface, 

whose  Equation  is  given  between  Three  Rectangular  Co- 
ordinates, X,  y,  z.     Examples 361-363 

269.  Moment  of  Inertia  of  Any  Solid.     Examples         ...     363,  364 

CHAPTER   XXXII 

Centre  of  Gravity.     Pressure  of  Fluids. 
Force  of  Attraction 

270,271.  Centre  of  Gravity.     Examples 365-369 

272,  273.  Tlieorems  of  Pappus.     Examples  ......  369,  370 

274.  Pressure  of  Liquids.     Examples 370-373 

275.  Centre  of  Pressure.     Examples 373-375 

276.  Attraction  at  a  Point.     Examples 375-377 

CHAPTER   XXXIII 

277.  Integrals  for  Reference 378-385 

Index     .............     386-388 


DIFFERENTIAL   CALCULUS 


CHAPTER  I 

FUNCTIONS 

1.  Variables  and  Constants.  A  quantity  which  may  assume  an 
unlimited  number  of  values  is  called  a  variable. 

A  quantity  whose  value  is  unchanged  is  called  a  constant. 
For  example,  in  the  equation  of  the  circle 

V  x'+f  =  a'', 

X  and  y  are  variables,  but  a  is  a  constant.  For  as  the  point  whose 
coordinates  are  x,  y,  moves  along  the  curve,  the  values  of  x  and  y 
are  continually  changing,  while  the  value  of  the  radius  a  remains 
unchanged. 

Constants  are  usually  denoted  by  the  first  letters  of  the  alphabet, 
a,  b,  c,  a,  /?,  y,  etc. 

Variables  are  usually  denoted  by  the  last  letters  of  the  alphabet, 
X,  y,  z,  <},,  i{/,  etc.  V 

2.  Function.  When  one  variable  quantity  so  depends  upon  an- 
other that  the  value  of  the  latter  determines  that  of  the  former,  the 
former  is  said  to  be  a  function  of  the  latter. 

For  example,  the  area  of  a  square  is  a  function  of  its  side ;  the 
volume  of  a  sphere  is  a  function  of  its  radius ;  the  sine,  cosine,  and 
tangent  are  functions  of  the  angle ;  the  expressions 


x\   logCr'  +  l),    Va;(a;  +  1), 
are  functions  of  x. 


J^,  \og{x^  +  y-z), 


2  DIFFERENTIAL   CALCULUS 

A  qiiantity  may  be  a  function  of  two  or  more  variables.  For 
example,  the  area  of  a  rectangle  is  a  function  of  two  adjacent  sides; 
either  side  of  a  riglit  triangle  is  a  function  of  the  two  other  sides ; 
the  volume  of  a  rectangular  parallelopiped  is  a  function  of  its  three 
dimensions. 

The  expressions 

x'  +  xy  +  if,   \o^{x'  +  y%   a^+^ 

are  functions  of  x  and  y. 
The  expressions 

xy  +  yz  +  zx, 
are  functions  of  x,  y,  and  z. 

3.  Dependent  and  Independent  Variables.  If  y  is  a  function  of  x, 
as  in  the  equations 

y  =  x-,    ?/  =  tan4a-,     ?/  =  e"4-l, 

X  is  called  the  iyidependent  variable,  and  y  the  dependent  variable. 

It  is  evident  that  when  y  is  a  function  of  x,  x  may  be  also  regarded 
as  a  function  of  y,  and  the  positions  of  dependent  and  independent 
variables  reversed.     Thus,  from  the  preceding  equations, 
x=V^,   cc^itan-i^/,   x  =  \og,{y-V). 

In  equations  involving  more  than  two  variables,  as 
2;4-a;  — 2/  =  0,   io-\-wz  +  zx-\-y  =  0, 
one  must  be  regarded  as  the  dependent  variable,  and  the  others  as 
independent  variables. 

4  Algebraic  and  Transcendental  Functions.  An  algebraic  function 
is  one  that  involves  only  a  finite  number  of  the  operations  of  addi- 
tion, subtraction,  multiplication,  division,  involution  and  evolution 
with  constant  exponents.*  All  other  functions  are  called  tra^iscen- 
devtal  functions.  Included  in  this  class  are  exponential,  loganthmic, 
trigonometric  or  circular,  and  inverse  trigonometric,  functions- 

I,-oTK.-The  term  "hyperbolic  functions"  is  applied  to  certain 
forms  of  exponential  functions. 

*  A  more  general  definition  of  Algebraic  Function  is,  a  function  whose  rela- 
tion to  the  variable  is  expressed  by  an  algebraic  equation. 


FUNCTIONS  ^ 

5.  Rational  Functions.  A  polynomial  involving  only  positive 
integral  powers  of  x,  is  called  an  inteyral  function  of  x;  as,  for 
I'xaniple,  2  +  x  -  4  r=  +  3  o^. 

A  rational  fraction  is  a  fraction  whose  numerator  and  denominator 
arc  integral  functions  of  the  variable ;  as,  for  example, 

X*  +  X-  —2x 

A  rational  function  of  x  is  an  algebraic  function  involving  no  frac- 
tional powers  of  x  or  of  any  function  of  x. 

The  most  general  form  of  such  a  fimction  is  the  sum  of  an  integral 
function  and  a  rational  fraction ;  as,  for  example, 

?,x'-2x 


2or  +  x-l+- 


2:c2  +  l 


6.  Explicit  and  Implicit  Functions.  When  one  quantity  is  ex- 
pressed directly  in  terms  of  another,  the  former  is  said  to  be  an 
explicit  function  of  the  latter. 

Por  example,  y  is  an  explicit  function  of  x  in  the  equations 

2/  =  X-  +  2  a-,  y  =  V.r'  + 1. 

When  the  relation  between  _?/  and  x  is  given  by  an  equation  con- 
taining these  quantities,  but  not  solved  with  reference  to  y,  y  is  said 
to  be  an  iinplicit  function  of  Xy  as  in  the  equations 

axy  -\-hx->rcy  +  d  =  Oy  ?/  +  log  ?/  =  x. 

Sometimes,  as  in  the  first  of  these  equations,  we  can  solve  the 
iquation  with  reference  to  y,  and  tlius  change  the  function  from 
implicit  to  explicit.     Thus  we  find  from  this  equation, 

hx  +  (1  • 

y= 3L^. 

ax  +  c 

7.  Single-valued   and    Many- valued    Functions.       In    the   e(i nation 

y  =  x'-2x, 
for  every  value  of  x,  there  isoone  and  only  one  value  of  y. 
Expressing  x  in  terms  of  y,  we  have 

x=l±  ^/y  +  l. 


4  DIFFERENTIAL   CALCULUS 

Here  each  value  of  y  determines  t.o  values  of  ..     In  the  f onue 

case,  w  is  a  single-valued  function  of  x.  ^ 

Inthelattercase,a;isaftoo-mh.ed/*mciionof2/. 

An  n-valued  function  of   a  variable  x  is  a  function  that  has 
values  corresponding  to  each  value  of  x.  ^.^ihi^ited  nui 

The  inverse  trigonometric  function,  tan-  x,  has  an  unlimited  nui 
ber  of  values  for  each  value  of  x. 

a    Notation   of   Functions.      The    symbols    n^^-f^^^^^^^, 
and  the  like,  are  used  to  denote  functions  of  x.     Thus  instead 
is  a  function  of  .t,"  we  may  write 

y  =  f{x),    or    y  =  <t>(x)- 

operation,  although  applied  to  different  quantities.     Thus  if 

f{x)=x'  +  5, 

then  m  =  f  +  '^^  /(a)  =  a^  +  5, 

y(a  +  l)=:(a+l)'^-h5  =  a^  +  2a  +  6, 

T       n  fi.o=p  PXTu-essions  f(  )  denotes  the  same  operation  as 
fined  by  (ir^hafi:  the  oVeLion  of  squaring  the  quantity 

^S::'?:;:";  .0.  variah.es  a.  exp.essedhyeo.:uas 
tween  the  variables.  ,  ,  o  .fi 

Thus  if  fix,y)  =  x-  +  ^^y-y^ 

then        /(a,.)  =  a^  +  3a^-^^-  /(.,  a)  =  ^^  +  3  .a-a^ 

/(3,  2)  =  3^ +  3-3.2 -2^  =  23.  f(a,0)  =  a\ 

If        <t>(x,y,z)  =  x^  +  yz-f  +  '^> 

then         <^(3,l,-l)  =  3.^  +  l(-l)-^-  +  2  =  2^-' 
^(a,?.,0)  =  «3-6^  +  2;       <^(0,0,0)=2. 


FUNCTIONS  ,-, 

9.    Inverse  Function.     If  //  is  a  given  functiou  of  x,  roprosfntcd  b-,- 

.'/  =  <l>i.->-)^ I  1 

and  if  fi'om  this  relation  we  express  x  in  terms  of  //,  so  that 

X  =  !/'(//), (•_') 

then  each  of  the  functions  </>  and  i/^  is  said  to  be  the  inverse  (tf  the 
other. 

For  example,  if  ?/  =  ^'  =  <i>(.-^), 

then  X  =a/^  =  </'('/)■ 

Here  i/^,  the  cube  root  function,    is    the    inverse    of   <f>,    the   cube 
function. 

If  y  =  «^  =  «^C0» 

then  X  =  log,,//  =  i/'(?/). 

Here  ij/,  the  logarithmic  function,  is  the  inverse  of  (f>,  the  expouential 
function.  _      ,^ 

Again,  suppose  2/ =-j ^  =  <^(-'^") (•^) 

V  —  2 

From  this  we  derive  x  =  ' =  \h()/) (h 

?/  +  1  ■ 

Here  i/'  as  defined  b}'-  (4)  is  the  inverse  of  <^  as  defined  by  <.">)• 
The  notation  ^~^  is  often  employed  for  the  inverse  function  of  </>. 

Thus,  if  ij  =  <f>{x),  X  =  4>-\y). 

If  y^m,  x=f-Hy)- 

The  student  is  already  familiar  with  this  notation  for  the  invei 
trigonometric  functions. 

If  y  =  sin  X,  x  =  sin"'  y. 

EXAMPLES 

1.    Given  2x' —  2 xy +  y- =  a'; 

change  y  from  an  implicit  to  an  explicit  function. 
Ans.  y  =  X  ±  Vet"  —  a;'-. 


DIFFERENTIAL   CALCULUS 

2.    Given  sin  (x  —  ?/)  =  m  sin  // ; 

change  y  from  an  iiu[)licit  to  an  explicit  function. 
,  J.     _i      sin  X 


m  -\-  cos  X 

3.  Given  f{x)  =  2.x'-?,x-  +  x  +  2\ 

find        /(I),  .r(i?),  .ro),  /(- 1),  /(O). 

Show  that  ./■  {x  +  1)  -f{x)  =  6  a--, 

/(a;  +  7i)  =f(x)  +  (G  .r  -  6 .«  +  l)/i  +  (6  x  -  3)/r  +  2  /i''. 

4.  Given  i^(.v)  =  (a-- 1)-; 

show  that      F(x  +  1)  -  i^(:i-  -  1)  =  8  a.-^. 

5.  Given /(■f)  =  ^'"^+-;""^   find /(O),  /(.r) +/(-a;). 
Show  that  ./•  (2.I-)  -./■(-  2  .T)  =  [/  (.r)]'^  -  [/  (-  x)]'. 

6.  If  <^  (^)  =  e^  >  (rt  +  ?>)  =  <^  (a)  ^  (b)- 
Show  tliat  the  same  relation  holds  for  the  function 

ij,(e)  =  cos  0  -f  V'^^  sin  6, 
giving   ^ (a  +  6)  =  lA  (a) </'('->)• 

o  X  —  6 
show  that  the  inverse  fiiuction  is  of  the  same  form, 

8.    If  cf)(x)  =  -"~^   ' ,    find  the  inverse  function  of  <i>, 
ax  —  c 

Compare  the  two  functions  when  b  =  c. 


9.    If  f(x)  =  log„(x+Vx'-l), 

show  that   /-i(a-)  =  ^il±-^. 

10.    If      /  (x,  y)  ^  a.r  +  2  hxy  +  c//^    find  f(l,  2) ,f(y,-x). 

Show  that 
f(x  +  h,y  +  k)  =f{x,  y)  +  2{ax  +  by) h  +  2 (bx  +  cy)  k  +f(h,  k). 


11.    Given 


FUNCTIONS 


(f>{m,  n) 


m  \n 


where  m,  n,  are  positive  integers;  show  that 
<f>  {m,  ?i  +  1)  +  <^  (m  + 1,  n)  =  <f>(m  +  l,n  +  1). 

t;  y,  -2; 
12.    Given  /  {x,  y,  z)  =  z,  ^x]  y  ; 

y,  z,  X 
show  that  f{y  V«,  z  +  .t,  o;  +^  =  2/(x,  ?/,  z). 


CHAPTER   II 
LIMIT.     INCREMENT.     DERIVATIVE 

10.  Limit.  When  the  successive  values  of  a  variable  x  approach 
nearer  and  nearer  a  fixed  value  a,  in  such  a  way  that  the  difference 
x  —  a  becomes  and  remains  as  small  as  we  please,  the  value  a  is 
called  the  limit  of  the  variable  x. 

The  student  is  supposed  to  be  already  somewhat  familiar  with  the 
meaning  of  this  term,  of  which  the  following  illustrations  may  be 
mentioned. 

The  limit  of  the  value  of  the  recurring  decimal  .3333  ...,  as  the 
number  of  decimal  places  is  indefinitely  increased,  is  i. 

The  limit  of  the  sum  of  the  series  1  +  i  +  ^  +  i  H — j  as  the  number 
of  terms  is  indefinitely  increased,  is  2. 

,jj.'!  ^3 

The  limit  of  the  fraction- ,  as  x  approaches  a,  is  3al 

X  —  a 

The  circle  is  the  limit  of  a  regulai-  polygon,  as  the  number  of  sides 
is  indefinitely  increased. 

The  limit  of  the  fraction  ^H^-,  as  $  approaches  zero,  is  1,  provided 
6  is  expressed  in  circular  measure. 

11.  Notation  of  Limit.  The  following  notation  will  be  used : 
"Lim^^a"  denotes  "The  limit,  as  x  approaches  a,  of." 

For  example,  Lim^.^^^— ^^^^^^^ — -  =  2. 

X-  —  ax 

Li m,^o  (2  a;'  -  hx  +  h')  =  2  x^. 

12.  Some  Special  Limits.  There  are  two  important  limits  required 
in  the  following  ijia|»ter. 

(a)  Lin)(,^—    -,    6  heing  in  circular  measure, 
u 

•8 


r  that  IS,   2  a  sin 


LIMIT  .. 

le  AOA'  =  2  6,  and  let  a  be  the  radius  of  the  arc  AC  A'. 
try,  ABA'<ACA';         • 


<2a^,  ^i^^<l. 


(1) 


a6 


Also  from  geometrjl,  .ICl'  <  ABA' ;  ^(^^  ^sla^jjC^X 

7  that  is,     2ke  <  2  ti  tan  6L   ^^-^>  B, 

^>o„.<,./.     ...     .(3) 

Hence  by  (1)  and  (2),   ^Hli    is  inter- 
d  * 

mediate  in  value  between  1  aud  cos  6. 
As  6  approaches  zero,  cos  6  a]>})roaches  1. 
sin  6      ^ 


Hence 


Lim, 


The  student  will  do  well  to  compare  the  correspondiiiL,'  vain 
and  sin  9,  taken  from  the  tables,  for  aiii^-les  of  i'°,  V,  and  Id 


Angle 

e 

siii^ 

1° 
10' 

—  =  .0872665 

-^  =  .0174533 

180 

^    =  .0029089 

JOSO 

.0871.")7 
.0174.~.24 
.0029089 

I 


(b)  Lim__^  [1  H ]_     IJefore  deriving  tliis  limit  let    us  coiiquit'' 

the  value  of  the  expression  for  increasing  values  of  z.     Tlius, 
(1  +  ^)^  =  2.25 
(1  +  ^y  =  2.48832 
(1  +  -1^)10  =  2.59374 
(1.01)'^  =  2.70481 
(1.001)>'^»  =  2.71692 
(1.0001)'**^  =  2.71815 
(1.00001)'*^'  =  2.71827 
(1.000001  )'«»^'  =  2.71828 


10  DIFFERENTIAL   CALCULUS 

The  required  limit  will  be  found  to  agree  to  five  decimals  with  the 
last  number,  2.71828. 

By  the  Binomial  Theorem, 


,'hich  nuiv  be  writUMi  i       /         ^\  f -i       2\ 


+ 


1  2 

When  z  increases,  the  fractious  -,  -  etc.,  approach  zero,  and  we 

,  z  z' 

have 

,    Li„,.„(l+^J  =  l+l+|  +  |  +  ^+-.- 

This  quantit}'  is  usuall}'  denoted  by  e,  so  that 

e=l+-+  -  +  -+-+•". 
1      [2  ^     |_4 

The  value  of  e  can  be  easily  calculated  to  any  desired  number  of 
decimals  by  computing  the  values  of  the  successive  terms  of  this 
series.     For  seven  decimal  places  the  calculation  is  as  follows: 

1. 

2)  1. 

3)  .5 


4) 

.106666667 

5) 

.041066667 

6) 

.008333333 

') 

.001388889 

8) 

•000198413 

9) 

.000024802 

10) 

.000002756 

11; 

.000000276 

.000000025 

e  =2.7 182818-.. 
This  quantity  e  is  the  base  of  the  Napierian  logarithms. 

*  For  a  rigorous  derivation  of  this  limit,  tlie  student  is  referred  to  more  ex- 
tensive treatises  on  the  Differential  Calculus. 


DERIVATIVE 


11 


13.  Increments.  An  increment  of  a  variable  quantity  is  any  ackli- 
tiou  to  its  value,  and  is  denoted  by  the  symbol  A  written  before  this 
quantity.    Thus  A.r  denotes  an  increment  of  x,  A//,  an  increment  of  >/. 

For  example,  if  we  have  given 

y  ^  A 
and  assume  a;  =  10,  then  if  we  increase  the  value  of  x  by  2,  the  valuf 
of  y  is  increased  from  100  to  144,  that  is,  by  44. 

In  other  words,  if  we  assume  the  increjuent  of  x  to  be  A.c  =  2,  w».' 
shall  lind  the  increment  of  y  to  be  Ay  =  44. 

If  an^increment  is  negative,  there  is  a  decrease  in  value. 
For  example^  calling  x  =  10  as  before,  in  y  =  .tt, 

if    A.«  =  -  2,  then     A//  =  -  36. 

14.  Derivative.     With  the  same  equation, 

y  =  ar, 
and  the  same  initial  value  of  x, 

X  =  10, 
let  us  calculate  the  values  of  A?/  corresponding  to  different  values  of 
Ax.     We  thus  find  results  as  in  the  following  table. 


If  A.«  = 

then  A//  = 

and^'^-=/i^^ 
A.r 

3. 

09. 

23.  .  2^7^:? 

o 

44. 

22.    ^^^  ^ 

1. 

21. 

21     ^^'' 

0.1 

2.01 

20.1  ^'"' 

0.01 

0.2001 

20.01  ^f 

0.001 

0.020001 

20.001 

7i 

20  h  + 7r 

20  +  /t 

The  third  column  gives  the  value  of  the  ratio  between  the  incre- 
ments of  X  and  of  y. 

It  appears  from  the  table  that,  as  Ax  diminishes  and  approaches 
zero.  Ay  also  diminishes  and  approaches  zero. 


12  DIFFERENTIAL   CALCULUS 

The   ratio    --''  diminislies,  but   instead  of   approaching  zero,  ap- 
A.<' 
preaches  20  as  its  limit. 

This  limit  of  —  is  called  the  derivative  of  y  with  respect  to  x, 
A.f 

and  is  denoted  by  ~.     In  this  case,  when  a;  =  10,  the  derivative 

dx 

It  will  be  noticed  tliat  the  value  20  depends  partly  on  the  func- 
tion y  =  XT,  and  partly  on  the  initial  value  10  assigned  to  x. 

Without  restricting  ourselves  to  an}'  one  initial  value,  we  may  ob- 
tain ^  from  ?/  =  X-. 
dx 

Increase  x  by  A.r.     Then  the  new  value  of  y  will  be 

y'  =  (x  +  Ax)-'; 
therefore.        A//  =  y'-  y  =  (x  +  A.r)-'  -  a--  ^  2x^x  +  (Axf. 

Dividing  by  A.e,        ^  ^  o  .}•  +  A.v. 

A.« 

The  limit  of  this,  when  A.i-  approaches  zero,  is  2x. 

Hence,  -^  =  2  x. 

dx 

The  clerivatjve  of  a  function  may  then  be  defined  as  the  limiting 
value  of  the  ratio  of  the  iiirrcnicnt  of  the  function  to  the  increment  of  the. 
variable,  as  the  latter  iiu-rement  aiyproaches  zero. 

It  is  to  be  noticed  that  -—  is  not  here  defined  as  b,  fraction,  but  as 
dx  . 

a  single  symbol  denoting  the  limit  of  the  fraction  — .     The  student 

Avill  find  as  he  advances  that  —  has  many  of  the  properties  of  an 
ordinary  fraction. 

Tlie  derivative  is  sometimes  called  the  differential  coefficient. 

15.   General  Expression  for  Derivative.     In  general,  let 

Increase  x  by  A.r,  and  we  have  the  new  value  of  y, 
y'=f(x  +  ^x). 


DERIVATIVE 


13 


^.'/  = .'/'  -  y  =./■(•'•  +  ^•^)  -./■(•«). 

A^  _f{x  +  A-r)  —fix) 
Aa;  Aa;  ' 

r/.f  AiB 

Geometrical   Illustration.     The   process  of  finding  the    derivative 
from  y  —  x^,  may  be  illustrated  by  a  square. 

Let  X  be  the  length  of  the  side  OP,  and  y  tlie  area  of  the  s(iu:ire 
on  OP. 

That  is,  y  is  the  number  of  square  units 
*    corresponding  to  the  linear  unit  of  x. 

When  the  side  is  increased  by  PP',  the 
area  is  increased  by  the  space  between  the 
squares. 

That  is,    A.v  =  2.rA.^-+(A.T)-,  ^^=2x  +  ^x, 


^/=Lim,_^=2l 
dx  A.r 


^i> 

y 

CC 

.r 

A  a- 

p.  p' 


16.  From  the  definition  of  the  derivative  we  have  the  following 
process  for  obtaining  it : 

((/)  Increase  x  by  A.i",  and  by  substituting  x  +  A.)"  for  a-,  deter- 
mine y  -f  A//,  the  n^W  value  of  y. 

(b)  Find  Ay  by  subtracting  the  initial  value  of  y  from  tlie  new 
value. 

(c)  Divide  by  Ax,  giving  —  • 

A.f 

(d)  Determine  the  limit  of  _  :',   as    A.r   ap})roaches   zero.       This 

Aa* 

limit  is  the  derivative  — . 
dx 
Apply  this  process  to  the  follov;ing  examples. 

EXAMPLES 
1.   y^2x'-6x-{-5. 

Increasing  x  by  Aa*,  we  have 

y  +  ^y=2(x  +  Axf  ~  6(x  +  Ax)  +  5; 

therefore,        Ay  =  2  (x  +  Aa-)"  -  6  (a-  +  Ax)  +  /)  -  2  r"  4-  <>  J"  -  •'"> 

=  (6  a--  -  G)  Aa-  +  G  x(Ax)-  +  2(A  xf. 


14 


DIFFERENTIAL   CALCULUS 


Dividing  bv  Ax, 


^  =  6x'-(>  +  (J  xAx  +  2  (Axy. 
Ax 

^  =  Lim^.^„^  =  6x^-6. 
dx  -^  "  Ax 


2. 


y  +  ^11 


Ay 


Ay 


'x  +  1 

X  +  Ax 
'  a;  +  :1a;  +  1 

X  +  Ax 


■  y 


Ax 


:  -^"Ax  +  1      x-\-l      {x  +  Ax  +  1)  (a;  +  1) 


Aa;  ^  (a;  +  Ax  +  l)(a;  +  1)' 

dy  _  -r  •  A?/  _       1 

di~     ^"'^^=°Aa;~(a;  +  l)^ 

?/  =  VaJl 


A' 


7/  +  A.v  =  Va;  +  Aa;, 

Ay  =  Va;  +  Aa;  —  Vx,  (j  ^    v/ 


-+ 


A?/ _  Vx-+  Ax—  -\/x_ 
Ax  Ax 

The   limit   of    this    takes   the    indeterminate  form   -.      But   b} 

rationalizing  the  numerator,  w% have 

Ay  ^ Ax 1 

^•^"      A.^•(  V;c  +  Ax-  +  Vx)      Va;  +  Ax-\-  Vx 


^=T-  Af/_     1 

da;  -^^^''A.*;      2-^3; 


^^^ 


4.  y  =  x*-2  X-  +  3  X  —  4, 

■  /     ••      '  - 

5.  y={x  —  af, 


dx- 

^  =  3(a;-a)^ 
da;        ^         ^ 


DERIVATIVE 

6. 

y=(t  +  2){3- 

-20, 

/  7- 

/ 

-^' 

/ 

8. 

^^a^'  ' 

.    -.■ 

9. 

y  +  ^ 

QlO. 

11. 
12. 

/ 

r            ^ 

(^-1/' 

y=V.f  +  2, 

, 

i    13. 

^  =  a;2, 

14. 

Z/  =  Va^  -  xf, 

15. 

"V 

'      "'^    • 

^^-4^-1. 
dt 


(/,// 

IU)I 

dx 

{ii-xf 

dx 

9 

d!/ 

C'/  +  2/ 

d!/_ 
dx 

:1         2a^ 
(x  +  a) 

dx_ 
dt~ 

t  +  1 

(t-lf 

^  = 

1 

(Ix     2Vx+2 


d}/  ^  Sx^' 
dx       2  ' 


fZ.v 


^■__  JL 

16.  Show  that  the  derivative  of  the  area  of  a  oirclo,  witli  n-spc. 
to  its  radius,  is  its  circumference. 

17.  Show  that  the  derivative  of  the  vohune  of  a  sphere,  will 
respect  to  its  radius,  is  the  surface  of  the  splierc. 

AVe  shall  now  give  sonip  illiis;fr;iti"n><  <■('  tlic  iii.'niuir'-  ,.l'  tli.-  ilt-riv.i 
tive. 


16 


DIFFERENTIAL   CALCULUS 


17.    Direction   of   a  Plane  Curve.     This  is  one  of  the  simplest  and 
most  useful  interpretations  of  the  derivative. 

Let  P  be  a  point  in  a  curve  determined  by  its  equation  ij—fix), 
and  FT  the  tangent  at  P. 

Let  OM==x,  MP  =  y. 

If  we  give  x  the  increment 
A.«  =  MN,  y  will  have  the  in- 
crement A?/,=  RQ. 

Draw  PQ,     llien 

-^-«  =  f^^.-  ■ 

Now    if    A.r    diminish 
aj)proach    zero,    A//   will 
approach    zero,    the    point    Q 
will    move    along    the    curve 

towards    P,  and  PQ  will  approach   in   direction  PT  as  its  limit 
Taking  the  limit  of  each  member  of  (1),  we  have 

tan  TPPi, 


Lim^.r=o-^^ 

Ax" 


dy 
dx 


at  any  point  of  a  curve,  is  the  trigono 


That  is,  the  derivative 

metric  tangent  of  the  in- 
clination to  OX  of  the 
tangent  line  at  that 
point. 

This  quantity  is  de- 
noted by  the  term  slojye. 

The  slope  of  a  straight 
line  is  the  tangent  of  its 
inclination  to  the  axis 
of  X. 

The  slope  of  a  plane 
curve  at  any  point  is  the 
slope  of  its  tangent  at 
that  point. 

Thus,  -^ ,  at  any  i)oint  of  a  curve,  is  the  slope  of  the  curve  at  that 
.    ,        dx 
point. 


derivativp:  17 

For  example,  consider  the  parabola  xr  =  4/)//,     y  =  — . 

The  slope  of  the  curve  is  ^  =  —  . 
dx      'Ip 

At  0,  where  x  =  0,  the  slope  —  0,  the  direction  bein;^  horizontal. 

At  L,  where  x  =  2p,  the  slope  =  1,  corresponding  to  an  inclination 
cu  45°  to  the  axis  of  X. 

Beyond  L  the  slope  increases  towards  co,  the  inclination  increasing 
towards  the  limit  90°. 

For  all  points  on  the  left  of  OY,  x  is  negative,  and  hence  the  slope 
is  negative,  the  corresponding  inclinations  to  the  axis  of  X  being 
negative. 

/  18.   Velocity  in  Terms  of  a  Variable  t  denoting  Time.     A  body  moves 
over  the  distance  OP  =  s  in  the  time  t,  s  being  a  function  of  /;  it  is 
required  to  express  the  velocity  at  the  point  P. 
Let  As  denote  -tU'--^' 

the        distance     !r ^ ^, 

PP    traversed 

in  the  interval  At.     If  the  velocity  were  uniform  during  this  interval, 

it  would  be  equal  to  ^ • 

For  a  variable  velocity,  --^  is  the  average  or  mean  velocity  between 

At  ■ 

P  and  P,  and  is  more  nearly  equal  to  the  velocity  at  P  the  less  w<' 
make  A^ 


That  is,  the  velocity  at  P  =  Lim  ^,^  "a/  ~"  r/« 


A.s      (Is 


If  V  denote  this  velocity,  v  = 

Thus,  —  is  the  rate  of  increase  of  s. 
cU 

Similarly,  ^  and  37  are  the  rates  of  increase  of  x  and  y  respect 
•"  ai  at 


18 


DIFFERENTIAL   CALCULUS 


19.    Acceleration.     The  rate  of  increase  of  the  velocity  v  is  called 

acceleration. 

If  we  denote  this  by  a,  we  have  by  the  preceding  article, 

(Iv 
«  =  — -• 

dt 

For  example,  suppose  a  body  moves  so  that 
■      Then  the  velocity. 


and  the  acceleration, 


dt 

do      p , 
«  =  —  =  6t. 
dt 


20.   Rates  of  Increase  of  Variables.     For  further  illustrations  of  the 

derivative,  consider  the  two  following  problems: 

Problem   1.       A   man 

N 


walks  across  the  street 
from  A  to  B  at  a  uniform 
rate  of  5  feet  per  second. 
A  lamp  at  L  throws  his 
shadow  upon  the  wall 
MK  AB  is  36  feet,  and 
BL  4  feet.  How  fast  is 
the  shadow  moving  when 
he  is  16  feet  from  yl? 
When  26  feet  ?  When 
30  feet? 

Let   P  and    Q   be   si- 
multaneous positions  of  man  and  shadow- 
y      BL  4.        ..         4  a; 

X 


.V^\ 

V 

\\ 

X  -, 

\>.^i 

!4      B 

'.G> 

p^ 

N 

Let  xiP=x,  AQ  =  y. 


Then 


y  = 


(1)1 

PB     36  -x'    "      36  - .);  ^ 

AVhen  he  walks  from  P  to  P,  the  shadoAv  moves  from  Q  to  Q'. 
That  IS,  when  ^x  =  PP\  Ay=QQ'. 

Let  A<  be  the  interval  of  time  corresponding  to  Ax  and  A?/. 
A// 

Then  we  may  write  a>~Ax*     *     °      "     *     "     *      '     *      ^^-^i 

Ai 


DERIVATIVE 


19 


Art.  IS. 


If  now  we  suppose  M  to  diminish  indefinitely,  Ay  and  Ax  will 
also  diminish  indefinitely,  and  we  have  for  the  limits  of  the  two 
members  of  (2),  , 

^_'JI__  i'«ate  of  increase  of  y 
dx     dx     rate  of  increase  of  x 
'dt 

rpj^^^  -g  velocity  of  shadow  at  any  i)oint  Q  _  d>/ 

velocity  of  man  ~  dx 

Finding  the  derivative  of  (1),  Ave  have 
d>/ 


144 


Hence, 


dx      Q)G  —  x)- 


144 


velocity  of  shadow  at  any  point  Q  = 

(:>()  — xf 
144 


See  V^i.  8,  \v^\6. 
(velocity   of  man) 


(oG  -  X) 
720 


(5  feet  per  second) 
,  feet  per  second 


I  (36  -  xy 

I  =    1.8  feet  per  second,  when  x  =  IG  ; 

I  =    7.2  feet  per  second,  when  x  =  2G  ; 

=    20  feet  per  second,  when  x  =  30. 

Problem  2.  The  top  of  a  ladder  20  feet  long  rests  again.st  a  wall. 
,The  foot  of  the  ladder  is  moved  away  from  the  wall  at  a  uniform 
rate  of  2  feet  per  second. 
How  fast  is  the  top  moving, 
when  the  foot  is  12  feet 
from  the  wall  ?  When  16 
feet  from  the  wall  ? 

Suppose    PQ  to   be    one 

position  of  the  ladder. 

Let 
7 

AP=x,  AQ  =  rj. 

Then 

y=^MO^^.     (3) 


20  DIFKKltKNTlAL    CALCULUS 

When  the  foot  moves  from  P  to  P',  the  top  moves  from   Q  to 
That  is,  if  ^x  =  PF,  Afj  =  QQ'. 
In  the  same  way  as  in  Problem  1, 


&id  from  this, 


that  is, 


Ax 

At 
~  Ax' 

aT 

dy 

dx  ' 

dt 

dt 

velocity  of  top  at  Q  _  dy 
velocity  of  foot  dx 


From  (3),  ^  -  -— =^=.  See  Ex.  14,  Art.   , 

^  ^'  dx     V400-:c2 

Hence, 

velocity  of  top  at  any  point  Q  = n=^  (velocity  of  foot) 

V400— x-^ 


x 

feet  per  second. 


V400- 

The  negative  sign  is  explained  by  noticing  from  the  figure  tha 
decreases  when  x  increases.  Hence  the  rates  of  increase  of  x  an( 
have  different  signs. 

AVhen  x  =  12,        velocity  of  top  =  —  l.V  feet  per  second. 

AVhen  x  =  16,        velocity  of  top  =  —  2|  feet  per  second. 

From  these  problems  it  appears  that,  while  -^  is  the  ratio  betwiji 

the  increments  of  ?/  and  x,  -r-  is  the  ratio  between  the  rates  of  ivcre 
dx  •   . 

of  these  variables. 


DERIVATIVE 


21 


21.    Increasing  and   Decreasing   Functions.     If  the  derivative   of  a 

functioH.  of  X  is  positive,  the  function  increases  when  x  iiicreaaes  ;  and  if 
the  derivative  is  negative,  the  function  decreases  tchen  xjncreases. 

For  if  the  derivative  ^,  which  is  the  ratio  between  the  rates  of 
-  dx- 

increase  of  the  variables  (see  conclusion  of  Art.  20),  is  positive,  it 
follows  that  these  rates  must  have  the  same  sign  ;  that  is,  y  increases 
when  X  increases,  and  decreases  when  x  decreases. 

But  if  -^^  is  negative,  the  rates  must  have  different  signs ;  that  is, 
y  decreases  when  x  increases,  and  increases  when  x  decreases. 

This  is  also  evident  geometrically  by  regarding  ^-^  as  the  slope  of 

dx 
a  curve. 

As  we  pass  from  Ato  B,  y  increases  as  x  increases,  but  from  B  to 
C,   y  decreases   as   x  in- 
creases. ^ 

Between  A  and  B  the 

slope  -^  is  positive  ;  be- 
tween B  and  C,  negative. 

Xcuthe^jformer  case  y  is 
said  to  be  an  increasing 
function ;  in  the  latter 
case,  a  decreasing  function. 

For    example,  consider 
the  function  y  =  a?,  from  which  we  find  ^  =  3af'. 
7  '^-^ 

Since  —  is  positive  for  all  values  of  x,  the  function  y  =  o^  is  an 
dx 
increasing  function. 

If  we  take  y  =  ~,  we  find  —  = -• 

X  dx  x- 

Here  we  have  a  decreasing  function  with  a  negative  derivative.^ 
Another  illustration  is  Ex.  1,  Art.  U), 

When  X  is  numerically  less  than  1,  y  is  a  decreasing  tunction. 
When  X  is  numerically  greater  than  1,  y  is  an  increasing  function. 


22 


DIFFERENTIAL   CALCULUS 


22.  Continuous  Function.  A  function,  y  =  f(x),  is  said  to  h 
coutinuous  for  a  certain  value  x^,  of  x,  when  yo  =  f  (a^o)  is  a  deiinit 
quantity,  and  A?/o  approaches  zero  as  Axq  approaches  zero,  Axq  bein 
positive  or  negative. 

The  latter  condition  is  sometimes  expressed,  "when  an  infinite! 
small  increment  in  x  produces  an  infinitely  small  increment  in  v/ 

The  most  common  case  of  discontinuity  of  the  elementary  function 
(algebraic,  exponential,  logarithmic,  trigonometric  and  inverse  trigc 
uometric,  functions)  is  when  the  function  is  infinite. 


Y 

a 

[^ 

-      — — --^        ° 

A 

For  example,  consider  the  function  ?/  = ,  which  is  continuous 

for  all  values  of  x  except  x  =  a.  '  ~ 

When  x  =  a,  y=  cc,  that  is,  ?/  can  be  made  as  great  as  we  please  by 
taking  x  sufficiently  near  a.  Also  when  x<a,  y  is  negative,  and 
when  a;>a,  y  is  positive. 

There  is  a  break  in  the  curve  when  x  =  a,  and  the  function  is  said 
to  be  discontimious  for  the  value  x  =  a. 


DERIVATIVi: 


23 


The  function 


is  likewise  discontinuous  wlicn  x  =  a. 


1 

-      {x-af 

This  function  being  positive  for  all  values  of  x,  the  two  brani-lus 
of  the  curve  are  above  the  axis  of  x. 

Likewise  the  functions,  tan  x,  sec  a-,  are  discontinuous  when  j-=  ^ 

In  general,  if/(.T)  =  x,  when  x  =  a,  there  is  a  break  in  the  curve- 

7/  =f{x)  corresponding  to  x  ^  a,  and  l)oth  the  curve  and  the  function 

are  then  discontinuous.  - 

2'-»-2 
Another  form  of  discontinuity  is  seen  in  the  function  y  =  — ; , 

when  a;  =  0.  2-1-1 

Here  y  approaches  two  limits,  according  as  x  approaches  zero 
through  positive  or  negative  values. 


Lim, 


2^+1 

0  the 


We  see  that  when 
curve  jumps  from  y=2  to  y=l, 
that  is  from  B  to  A. 

The  function   is  discontinu- 
ous for  X  =  0. 

It  is  to  be  noticed  that  the 
definition    of    the    derivative 


24 


DIFFERENTIAL   CALCULUS 


implies  the  continuity  of  the  function.     For  —^  cannot  approach  a 

limit,  unless  A?/  approaches  zero  when  Aa;  approaches  zero. 

The  converse  is  not  true.  There  are  continuous  functions  which 
have  no  derivative,  but  they  are  never  met  with  in  ordinary 
practice. 

EXAMPLES 


J<.    1.   The  equation  of  a  curve  is  y  =  '——  ar  +  2. 

(a)     Find  its   inclination  to  the  axis  of  x,  when  x  =  0,   and 


when  x=l. 

(b)  Find     the 
points      where      the 
curve  is    parallel   to 
the  axis  of  X. 
Ahs.   x=0  and  x=2. 

(c)  Find  the 
points  where  the 
slope  is  unity. 

Ans.    x  =  (l±^2). 

(d)  Find  the 
point  where  the  direc- 
tion is  the  same  as 
that  at  x  =  3. 


Ans.     0°  and  135° 


A71S.   x  =  —  l. 


2.   In  Problem  1,  Art.  20,  when  will  the  velocity  of  the  shadow 
be  the  same  as  that  of  the  man  ?  Ans.    When  AP=  24:  ft. 

When  one  quarter,  and  when  nine  times,  that  of  the  man  ? 

Ans.   When  AP  =  12  ft.,  and  32  ft. 


3.  A  circular  metal  plate,  radius  r  inches,  is  expanded  by  heat, 
the  radius  being  expanded  m  inches  per  second.  At  what  rate  is 
the  area  expanded  ?  Ans.    2  -n-rm  sq.  in.  per  sec. 


/GO 


4.   At  what  rate  is  the  volume  of  a  sphere  increasing  under  the 
conditions  of  Ex.  3  ?  Ans.   4  wi^m  cu.  in.  per  sec. 


DERIVATIVK  Of) 

5.  The  radius  of  a  spherical  soap  bubble  is  increasing  uniformlv 
at  the  rate  of  j\  inch  per  second.  Find  the  rate  at  which  tli'e 
volume  is  increasing  Avhen  the  diameter  is  3  inches. 

Ans.     ?^  =  2.827  cu.  in.  per  .s.r. 

_^      6.    In  Exs.  5, 7,  Art.  16,  is  y  an  increasing  or  a  decreasing  fum-tinn  .' 

Is  '''*'  an  increasing  or  a  decreasing  function  of  x.' 

x  +  1 

7.  In  the  Example  1,  above,  for  wliat  values  of  x  is  .»/  an  increas- 
ing function  of  x,  and  for  what  values  a  decreasing  function  '.' 

„i^  8.  Find  where  the  rate  of  change  of  the  ordinate  of  the  curve 
?/  =  a-^— 6ar^4-3.«-f  5,  is  equal  to  the  rate  of  change  of  the  slope  of 
the  curve.     Ans.    a;=  5  or  1. 

■  9.    When  is  the  fraction  — '- increasing  at  the  same  rate  as  x! 

X-  -f  a- 

Ans.  When  ar  =  a'. 

10.  If  a  body  fall  freely  from  rest  in  a  vacuum,  the  distance 
through  which  it  falls  is  approximately  s  =  16  f;  where  «  is  in  feet, 
and  t  in  seconds.  Find  the  velocity  and  acceleration.  What  is  the 
velocity  after  1  second?     After  4  seconds  ?     After  10  seconds  ? 

Ans.  32,  128,  and  320  ft.  per  see. 


CHAPTER   III 
DIFFERENTIATION 

23.  The  process  of  finding  the  derivative  of  a  given  function  is 
called  (lifferevtiation.  The  examples  in  the  preceding  chapter  illus- 
trate the  aneauing  of  the  derivative,  but  the  elementary  method  of 
differentiiition  there  used  becomes  very  laborious  for  any  but  the 
simplest  functions. 

Differentiation  is  more  readily  performed  by  means  of  certain 
general  rules  or  formula!  expressing  the  derivatives  of  the  standard 
functions. 

In  these  formula?  u  and  v  will  denote  variable  quantities,  func- 
tions of  x;  and  c  and  n  constant  quantities. 

It  is  frequently  convenient  to  write  the  derivative  of  a  quantity  ?«, 

d  •  ,  1  p  du 
—  u  instead  ot  — , 
dx  dx 

the  symbol  —  denoting  "derivative  of." 

^  dx 

Tlius  Li-LlLU  the  derivative  of  (k.  +  v),  may  be  written  --  iu  +  v). 

2A.    Formulae  for  Differentiation  of  Algebraic  Functions. 

L     ^=1. 
dx 

II.     —  =  0.  ' 

dx 

III.    iL (,,  +  „)  =  ^*  +  ^. 

dx  dx     dx 


DIFFERENTIATION 


TAT         d    ,      ..  (lu    ,       (Ir 

dx  cLc        dx 

-vr        f^  /     N         da 
V.     — (cu)  =  c  — 
dx^    '        dx 

du        dv 

V M 


dx\vj  V- 

These  formulae  express  the  following  general  rules  of  differcnti- 
atiou : 

I.      Tlie  derivative  of  a  variable  icith  respect  to  itself  in  uniftf. 
II.      TJie  derivative  of  a  constant  is  zero^ 

III.  TJie  derivative  of  the  sum  of  two  variables  is  the  sum  tf  fhfir 
derivatives. 

IV.  The  derivative  of  the  product  of  tioo  variables  is  the  sum  of 
the  products  of  each  variable  by  the  derivative  of  the  otiiek 

V.     TJie  derivative  of  the  product  of  a  constant  and  a  variable  is 
the  product  of  the  coyistant  and  the  derivative  of  the  variable. 

VI.  TJie  derivative  of  a  fraction  is  tJie  derivative  of  tJie  nnmeraior 
midtiplied  by  tJie  denominator  minus  tJie  derivative  of  tJie  denotni- 
nator  midtiplied  by  tJie  numerator,  tJiis  difference  being  divided  by  the 
square  of  tJie  denominator. 

VII.  The  derivative  of  any  poiver  of  a  variable  is  tJie  product  of  the 
exponent,  tJie  power  witJi  exponent  diminisJied  by  1,  awd  tJie  derivtUive 
of  the  variable. 

25.  Proof  of  I.    'This  follows  innnediunix   iiuui  the  definitiu!.  nf 

a  derivative.     For,  since  —  =  1,  its  limit  -r^  =  1- 
'  Ax  dx 

26.  Proof  of  II.  A  constant  is  a  quantity  wliuse  value  does  not 
vary.  *  , 

Hence  Ac  =  0  and  —  =  0 ;    therefore  its  limit  ""  =  0. 
Ax  dx 


28  DIFFERENTIAL   CALCULUS 

27.    Proof   of   III.     Let  y  —  u  +  v,  and  suppose  that   when  x  re-        i 
ceives  the  increment  Ax,  u  and  v  receive  th,e  increments  An  and  Av, 
respectively.     Then  the  new  value  of  y, 

y  +  Ay  =  u  +  Au  +  f  +  Av, 

therefore  Ay  =  An  +  Av. 

Divide  by  Ax ;  then 

Ay  _  Au      Av 

Ax      Ax      Ax' 

Now  suppose  Ax  to  diminish  and  approach  zero,  and  we  have  for 
the  limits  of  these  fractions, 

dy  _  clu      dv 
dx      dx      dx 

If  in  this  we  substitute  for  y,  u  +v,  we  have 

r?  .     ,     .      du  ,  dv 
dxr  '      dx      dx 

It  is  evident  that  the  same  proof  would  apply  to  any  number  of 
terms  connected  by  plus  or  minus  signs.     We  should  then  have 

— (h  ±'o±w±  •••)=  —  ±  —  ±  -—  ±  •••. 
dx  dx     dx      dx 

28    Proof  of  IV.     Let  y  =  \w ; 

then  y  +  A^  =  {n  -f-  Ai(){v  +  At'), 

and  Ay  =  (m  +  A»)(y  +  Ai')  —  uv  =  vAii  +  («  +  Ai<)A?;. 

Divide  by  Aa;; 

,,  A'/         A?t  ,  ,     ,    .    sA-y 

then  -^  =  ^) [-(?(  + Ait)  — . 

Ax        Ax  Ax 

Now  suppose  Ax  to  approach  zero,  and,  noticing  that  the  limit  of 
u  +  Alt  is  tc,  we  have 

dri        du  ,      dv 

-^  =  V \-u  — ; 

dx        dx         dx 

, ,    ,  .  (^  /     \       di(  ,      dv 

that  IS,  —  ( tcv) —V \-u — . 

dx  dx        dx 


DIFFERKX'I'IA'I'ION  29 

29.    Product  of  Several  Factors.     Formula  IV,  may  be  extended  to 
the  product  of  three  or  more  factors.     Thus  we  have 


-  (uviv)  =-^(av-  w)  =w  —  (uv)  +u 

dx^       ^     dx^         ^        dx      '         dx 


dx        dxj  I 


„du   ,        dr  ,        din 

=vw —  -j-  uw (-  nr  — . 

dx  dx  dx 


It  appears  from  the  preceding  that  the  derivative  of  the  product 
of  two  or  three  factors  may  be  obtained  by  multiplying  the  deriva- 
tive of  each  factor  by  all  the  others  and  adding  the  results. 

This  rule  applies  to  the  product  of  any  number  of  factors.  To 
prove  this,  we  assume 

d  /  \  dn,  ,  du.,  ,         ,  (/' 

dx\  J        '  dx  dx  II 

Then  —[  u,Uo  •••  njii„.,\  =  t(„+i  —  (  u,nn  •••  ti„]  +  n^n.,  ■•■  «„-"•'-*-' 
dx\      '  J  dx\      '  )  '  dx 

du,   ,  dn.,  ,  ,  du. 

dx  dx  dx 

dx 

Thus  it  appears  that  if  the  rule  applies  to  n  factors,  it  holds  also  for 
71  +  1  factors,  and  is  consequently  applicable  to  any  number  of 
factors. 

2746  derivative  of  the  prodvct  of  any  number  of  factor.^  is  the  attm  of 
the  products  obtained  by  multiplying  the  derivative  oj  each  factor  by  aU 
the  other  factors. 


30  DIFFERENTIAL   CALCULUS 

30.    Proof  of  V.     This  is  a  special  case  of  IV.,  —  being  zero.     But 

at 

we  may  derive  it  independently  thus : 
y^cu, 
y  -{-  Ay  =  c(u  +  An), 
Ay  =  cAu, 


Ay  _    Au 
Aa;        Aa;' 


dy       du  rf  /    \        du 

dx       dx  dx  dx 


31.    Proof  of  VI 

Let  2/  =  ^.       ■ 

V 

Then 

,    .         ?<  4-  Art 
y  +  Ay  =     ^        ; 

v+Av 

therefore 

.         u-{-Au      u      vAu~vAv 
V  +  Av      V       (y  +  Avyo ' 

and 

Ay        Ax         Ax 

Ax"   (y  +  Av)v 

Now  suppose  Aa;  to  approach  zero,  and  noticing  that  the  limit  of 
-u  +  Av  is  V,  we  have 

du        dv 

V u 

dy  _    dx        dx 

dx~        V' 
Or  we  may  derive  VI.  from  IV.  thus : 
Since  2/  =  -> 

V 

therefore  yv  =  u. 


DIFFERENTIATION  SI 

By  IV.,  4'  +  4"  =  fL', 

ax       dx     dx 

ydy_du_udv^ 
dx     dx     V  dx* 

du        dv 

V XI  — 

therefore  dy  _    ^^        ^^ 

dx  V- 

/  32.    Proof  of  VII.    First,  suppose  n  to  be  a  positive  integer. 

Let  y  =  u"^ 

then  y  +  A//  =  (u  +  An)", 

and  Ay  =  («  +  A«)"  —  u". 

Putting  u'  for  u  +  Ah,  we  have 

Ay  =  m'"  —  w"  =  (m'  —  ii)  (m'"-!  +  m'"-2u  4-  m'—^w-  +  •••  +  u"-'), 
that  is,  Ay  =  Au  (u'"-'^  +  ?t'"-2  m  +  ic"'-hr \-  n" "  >), 

^  =  («"'-i  +«"'-2w  +  „".-3 „-^ ...  4.  „'.-h  ^*. 

Aa;      ^  "^  Ax 

Now  let  Ax  diminish;  then,  u  being  the  limit  of  11',  eacli  of  the  / 
terms  within  the  parenthesis  becomes  w""*;  therefore 

du  ,  dt( 

-1-  =  nu"-^  —  • 
dx  dx 

Or  it  may  be  proved  by  regarding  this  as  a  special  case  of  Art.  2'.'. 
where  Wj,  w,?  •••  and  «„  are  each  equal  to  u. 

Then  —  («")  =»'  ' 1-« 1-  •••  to  ;t  terms 

dx  dx  dx 

n-\du 
=nu"^ — 
dx 

Second,  suppose  w  to  be  a  positive  fraction,  ^. 

p 
Let  y  =  "', 

then  y  =  u^ ; 

therefore  -1  (^Z')  =  ^  (u"). 

dx  '  ux 


32  DIFFERENTIAL  CALCULUS 

But  we  have  already  shown  VII.  to  be  true  when  the  exponent  is 
a  positive  integer;  hence  we  may  apply  it  to  each  member  of  this 
equation.     This  gives 

^^      dX'    ^        dx 

^,        r  dy     J)  xi^'^du 

therefore  -t'— ;  t^- 

dx      q  y''    dx 

p 
Substituting  for  y,  u'>,  gives 

dy  _  p  >(^~^  dn  _  p    j~'  du 

dx      q    p_pdx      q        dx' 
u    , 

which  shows  VII.  to  be  true  in  this  case  also. 


Third,  suppose  n  to  be  negative  and  equal  to  —  m. 
Let  .  y=u-'^  =  — ; 


(ir)       —  mu" 

1     ITT  dy         dx  dx  .       ^  ,du 

by  VI.,  -^  = ; = =  —  mir""^  -J-  • 

dx  tr'"  u-'"  dx 


Hence,  VII.  is  true  in  this  case  also. 


EXAMPLES 

Differentiate  the  following  functions  : 
1.    y  =  x*. 

'^  =  ±(x^). 
dx     dx 

If  we  apply  VII.,  substituting  u  =  x  and  n  =  4,  we  have 

dec     ^  dx  ^ 

•  Hence,  -^  =  4.c^. 

dx 


DIFFERENTIATION 
2.    y  =  3  a;^  +  4  a^. 

by   III.,    making  xi  =3.c^and  v  =  \x^. 

£(3..)=3£(.<),    „yV. 
=  3-4ar''=12r'. 
Similarly,  /  (4ar'')  =4-J^(af)  =4  •  3.r-  =  12.^2. 


da; 


Hence,  ^_  =  12  x-^  +  12  a;-  =  12  (a;^  +  sF), 


3.    2/  =  a;2  +  2. 


dx     dx  dx 


|(a,t)=|.-^,     by  VII 

|(2)=0,      by  11. 

Hence, 

dx     2 

4.   y=3Vx- 

2        1 

Va;     •*' 

da;     da;-       ^      da;^         ^      dar^      ^      da; 

__3_      J.  _  3 
2a;i      x^      ^ 


32       34  DIFFERENTIAL    CALCULUS 

a  i:  ^"  +  "^ 

dx      dx\x^  +  3  ' 

Applying  VI.,  making 
th  ti  —  x-'r  3  and  v  =  x-  +  3,  we  have 

d  fx  +  3\      ("^^  +  ^)|(-''^  +  •^)  -  (-^  +3)|(^"  +  3) 


dx\x-  +  3y  (a;-  +  3)=^ 

^a^  +  3-(x  +  3)2x^3-6x-x^ 
{x'  +  3y  (a^  +  3y    ' 

Hence,  dy^3-Gx-x\ 

dx         (.r^  +  3)- 

6.   2/  =  (.T^  +  2)l 

If  we  apply  VII.,  making 

2 

u  =  .^•^  +  2  and  n  =  - ,  we  have 

=  |(a.^  +  2)-*2x  =  — ^^. 
Hence,  -■ 


dx    ,3(a;-  +  2)3* 


7.    y  =  {x^  +  l)  V^^^^. 


I=£k^+^)(^-^)*]- 


DIFFEKENTIATIOX  3^ 

If  we  apply  IV.,  making 

u  =  af-\^l  and  v  =  (jt"  -  x)^-,  we  have 
£[(a-=  +  l)(.r^-a-)^] 

Hence  ^  =  ^ (.t^  +  1)  (3  x'-l)  (x'  -  a-) " ^  +  (^-^  -  x) ^-2x 

^  (ar  +  1)  (3  x'-l)+4 x(af-x)^7x'-2x'-  1 

2(cc3_a;)i  2(x'-x)' 

*      8.   y  =  3  x''  -  2  x''  +  a,"  -  5,  --^  =  3(10  or'  _  4  .r  +  •'- )  • 


10.    i/  =  (.T  +  2a)(x-a)^  g  =  3(.7r-a'0. 


11.   y=(.v^-a^y, 


dx  3  a;! 


Differentiate  Example  11  also  after  expanding. 


12    .._-^--  ^^ 


(ic  - 1)-  dx         (x  -  ly 

13.  ,  =  .(=^  +  5)*,  |  =  o(x»  +  l)(>'  +  5)». 


36 

I 

14. 

X 

"      Va=-^' 

15. 

.=  (^'-''^', 

'  16. 

^^-^- 

DIFFERENTIAL   CALCULUS 


dy a^ 


dX         (^^(2  _  a;2)l 


dx              X* 
dy_ a 


^^•^         2  X  Vox  —  x^ 
Differentiate  both  members  of  the  identical  equations,  Exs.  17-1 

1 7.    (x-  +  ax  -f-  o-)(x-  —  ax  +  a-)  =  x*  +  a2.^•-  +  a*. 


^  2  o;^'  -  3  a;  +  1  x-1      2  x-  -  1 


20.    X  =  t(f-+  a')'T,  ^  =  {nf"  +  a2)(«2  _^  ^2^,^. 

21  y^C^t'-^f  dy_6(3t'+4t)(2t'-3Y 

'  ^     (t^  +  2y '  dt  (f  +  2y 

22  y=t(2^1^  dy  _  2  aa?"  (24  a;  +  5  a)(a  +  2  .a^)' 

23.  3/=(a;  +  l)3(3a;-8)Xa;  +  2)«, 

^  =  3  (13  X-  -  24)  (a;  + 1)^3  x  -  8)\x  +  2) 
ax 

n-i  dy  _n(a;"4-l) 

24.  y  =  x(x^  +  n)->  d.-(,n  +  ,)l' 


25.   v  = 


dy 


^2ax-x^  dx      (2ax-af)i 


DIFFERENTIATIOX 

26.   y  =  (^-^¥,  <!}[__   ^{x-x^^ 

^*      '  dx  3  a;*      * 


27.  ,-2-^+ivr  .^, 

'/.'/  _            3 

iix       ,1^/,  r^ 

28.  x=(t^-2)^t\^^^ 

29.  y-^-'-a')\ 

dt     4(^^+1)5' 
c7.v  _             2  aV 

{x'  +  ce)^ 

f7-«      (.t''+rt-')*(a-'-a')5 

30.  2/  =  J^'-^'  +  l, 

31.  3/-6ar^4-6.r  +  l^ 

rf.  V  _                 ar'  -  1 

^^*-      (ar+a;+  1)  Vx*  +  x-^  +  1 
cly  _      12  0^2 

(4.r  +  l)^  fto      (4a.+  i)! 

32.    ?/  =  (x-  -  3  aa-)^(4  .r  +  8  ax  +  la  a-  )5^ 


rf^/^ 4(2j'''-9a»)    


33.  ?/=(.x-+ Va-2  +  l)"(»Va;-+l-a-), 

y^  =  (u-  -  Df  .<•  +  \  .»•-  a-  1  »". 
cZ.f      ^ 

34.  For  what  values  of  x  is  3a;''  — 8. r*  an  increasing  or  a  (iocn 
ing  function  of  .r  ?  i^  '     -J 

Ans.    Increasing,  when  x  >  2;    decreasing,  wlien 

35.  A  vessel  in  the  form  of  an  inverted  circular  o(»ne  ot   seim- 
vertical  angle  30°,  is  being  filled  witli   water  at  the  uniform  rate  of 
one  cubic  foot  per  minute.     At  what  rate  is  the  surface  of  the  wai- 
rising  when  the  depth  is  6  inches  ?  when  1  foot  ?  when  2  feet  ? 

Ans.    .7(5  in. ;  .19  in. ;  .0.")  in.,  jier  s' 


38  DIFFERENTIAL  CALCULUS 

36.  The  side  of  an  eriiulateval  triangle  is  increasing  at  the  rr 
of  10  feet  per  minute,  and  the  area  at  the  rate  of  10  square  feet  i 
second.     How  large  is  the  triangle?  Ans.   Side  =  69.28 

,■  37.  A  vessel  is  sailing  due  north  20  miles  per  hour.  Anoth 
vessel",  40  miles  north  of  the  first,  is  sailing  due  east  15  miles  p 
hour.  At  what  -rate  are  they  aj)proaching  each  other  after  o: 
hour?  After  2  hours?,  jhis.  Approaching  7  mi.  per  hr. ;  separatii 
15  mi.  per  hr, 

When  will  they  cease  to  approach  each  other,  and  what  is  th( 
their  distance  apart  ? 

Ans.    After  1  hr.  16  min.  48  sec.     Distance  =  24  m 

38.  A  train  starts  at  noon  from  Boston,  moving  west,  its  motio 
being  represented  by  s  =  9  f.  From  Worcester,  forty  miles  west  c 
Boston,  another  train  starts  at  the  same  time,  moving  in  the  sam 
direction,  its  motion  represented  by  s'  =  2  t^.  The  quantities 
are  in  miles,  and  t  in  hours.  When  will  the  trains  be  nearest  tc 
gether,  and  what  is  then  their  distance  apart  ? 

Ans.  3  P.M.,  and  13  mi 
When  will  the  accelerations  be  equal  ? 

Ans.    1  hr.  30  min.,  p.m 

39.  If  a  point  moves  so  that  s  —  V^,  show  that  the  acceleratior 
is  negative  and  proportional  to  the  cube  of  the  velocity.  How  is 
the  sign  of  the  acceleration  interpreted  ? 

40.  Given  s  =  --{-bt-;      find  the  velocity  and  acceleration. 

41.  A  body  starts  from  the  origin,  and  moves  so  that  in  t  seconds 
the  coordinates  of  its  position  are 

x  =  t^  +  4:t--'St,  y  =  ^-.3f-4t. 

^3 
Find  the  rates  of  increase  of  x  and  y. 

Also  find  the  velocity  in  its  path,  which  is 

|  =  x/(^]+(^)  •  Ans.  6f  +  5. 


V(fT-(IJ 


DIFFEUKNTIATION  ;,;. 

42.  Two  bodies  move,  one  on  the  axis  of  A',  and  tho  otlicr  ....  tl.n 
axis  of  Y,  and  in-;  niinntes  their  distances  from  the  orijjin  :i 

x  =  2f-6t  feet,    and  ?/  =  6  f  -  9  feet. 

,    At  what  rate  are  they  approaching  each  otlier  or  separating,  after 
1  minute  ?     After  3  minutes  ? 

Ans.    Approaching  2  ft.  per  min. ;  .separating  0  ft.  per  niin. 
When  will  thej  be  nearest  together  ?      Ans.   After  1  min.  .30  .sec. 

43.  In  the  triangle  ABC,  L  and  J/ are  the  jniddle  points  of  liC 
and  CA  respectively.  A  man  walkjs  along  the  median  AL  at  a  uni- 
form rate.  A  lamp  at  B  casts  his  shadow  on  the  .side  AC  Sliow 
that  the  velocities  of  the  shadow  at  A,  M,  C,  are  as  2-' :  :)■:  -l'^;  and 
that  the  accelerations  at  these  points  are  as  2^ :  ,*>•':  4'^ 

[  ,         SuGGESTiox.  —  P  being  any  position  of  the  man,  draw  from  L  a 
line  parallel  to  BP. 

33.   Formulae    for    Differentiation  of    Logarithmic    and    Exponential 

Functions. 

,  du 

VIII.     — log„M  =  log,e— . 
dx  ■     u 

du 

TA.'       d  1  da; 

IX.     —\og^u=  — 
dx  u 


X.     —  a"  =  log.  a.  a"  —  - 
dx  dx 


XI.     |e"  =  .«^^ 
dx  dx 


XII.     -^H''  =  yt('-''^"  +  loir,  n  ■  '/•'l- 
dx  '/'• 


40  DIFFERENTIAL   CALCULUS 

^  34.  Proof  of  VIII.     Let?/  =  logv", 

then                               y  +  A?/  =  log„  (it  +  A«), 
q  A?/  =  \og^(u  +  A»)  —  log„H  =  log„  — ^t 

\^  XI  J        u  \  'f  / 

;h  Dividing  by  Ace, 

„  Am 

^=io./i+^^ir^ (1 

Aa;    •         \  u  J      u 

If  Aaf* approach  zero,  Au  likewise  approaches  zero. 


Now    Lim^„^(  1  +  i^'V"  =  Lim,_  (l  +  ^ 


M   /  V  2 


For,  if  we  put  —  =  z, 

Alt 


and  as  Lu  approaches  zero,  z  approaches  infinity. 
But  in  Art.  12  we  have  found 

Lini^^  f  1  +  -  j  =  e ; 

therefore     Lim^„^  (l-\ — ^  r"  =  e. 

Hence,  if  we  take  the  limit  of  each  member  of  (1), 

du 

dy     -, ^dx 

-^  =  log„e_. 
ax  u 


DIFFERENTIATION  1 1 

35.  Proof  of  IX.     This  is  a  special  case  of  VIII.,  when  a  =  e. 

In  this  ease 

.log„e  =  kg„('=l. 

Note.— Logarithms  to  base  e  are  called  Xajnerian  logarithms. 
Hereafter,  when  no  base  is  specified,  Napierian  logarithms  are  to  be 
understood;  that  is, 

log  u  denotes  log,  w. 

36.  Proof  of  X. 

Let.  11  =  a". 

Taking  the  logarithm  of  each  member,  we  have 
log  y=n  log  a ; 

'^         I 
therefore  by  IX.,  dx      ,  ^     dn 

^         '  _  =  logo— . 

1/  «•'-' 

Multiplying  by  y  =  a",  we  have 

dy     1  ,,du 

dx  dx 

37.  Proof  of  XI.     This  is  a  special  case  of  X.,  where  a  =  e. 

0  38.   Proof  of  XII.     Let  ?/  =  ?<". 

Taking  the  logarithm  of  each  member,  we  have 
\ogy=v\ogu; 

dji      ^h^  ! 

tneref ore  by  IX.,  dx  _    dx     ■,  „^/]j^ 

1/         u  ^    dx 

Multiplying  by  ?/  =  u",  we  have 

dv  ,   i<^^"  ,  1  /''' 

ll^  =  r»'-'--  +  log  »•'/•-- • 
dx  dx  dx 

The  method  of  proving  X.  and  XII.  by  taking  the  logarithm  of 
each  member,  may  be  api)lied  to  IV.,  VI.,  and  VII. 
This  exercise  is  left  to  the  student. 


42  DIFFERENTIAL   CALCULUS 

1.  y  =  log  {2x^  +  3  x^), 

2.  y^  x"  log  (ax  +  6),  ^  =  x"  M  -^^^+w  log  (ax+b 

dx  [ax+b 

3    7/  —      1  ^IlL  —  —  1  +  ^og  -^ 

a;  log  .6- '  dx  (x  log  a;)"^ 

au>      3  a; +  2      o.<;+2 

c  T      a.T  —  5  c7i/         2  ab 

5.    ■?/  =  loe'  ■-  — 


EXAMPLES 

(See  note, 

Art.  35 

0 

> 

dx 

6 

2 

x^  +  3x 

dy_ 
dx 

X' 

^  ax  +  6  da;      a'-x-  —  b'- 


c  1     3  a;  4-1  dy  8 

6.    ?/ =  log — ,  -^  = 

x+o  dx     3  a;- +  10 a; 4- 3 

^  ^t'  +  t  +  l'  dt      ^^+^'  +  1 

8.  y  =  a'^e^,  ^  =  (1  +  log  a)  a^e'. 

dx 

n'      *    1      /  r  ,  ;x\  dii     a'' log  a  4- 5"=  locT  ?> 

9.  2/ =  log  (cr  4- ^0.  y^  = "   ,  ,  ,,      '      • 

a.i!  a""  +  b'' 


^^  =  8  e2-rfi2-  _ 
dx 

Differentiate  Ex.  10  also  after  expanding. 


10.  y  -  (f'  - 1)^  -^  =  8  e^H^-^  - 1)^ 

dx 


11  5  a;  4-  3  _o,  rZ?/      24  a-  - 10  x-  _2^ 

a;  — 3  dx         (^x  —  oy 

12.    v/  =.  (3  X  -  Vfe^-\  "^-^  =  3  (9  ar  -  l)e"-  =. 


DIFEERENTIATJOX  ^ 

''    '^  =  '''■'  S=^'S'(5  +  ...,og5). 

14.  ?/  =  loglogx 1_,  £^  _  1  +  log  X 

log  a;  (/.c     a-(logu;)-' 

Differentiate  both  luenibcrs  of  the  identical  equations,  Exs.  l.>-18. 

15.  {x  +  e'Y  =  a;-*  +  4  are'  +  6  ore'-^  +  4  xe^  +  e^. 

16.  (a'  —  e^)'''  =  a^  -  3  ci^e^  +  3  a^e-'  -  e^. 

17.  log  (e-'-  +  e-")  =  log  (e^-«  +  e"-')  +  .<•  +  a. 

18.  .t^^Sa^filogar^ 


l^^'-io..A.4-iv  ^y-  log^ 

(Zx     (x-  + 1)- 


19.  y  =  :^pji_log(:f+l) 

20.  y  =  log  ( V^rr^  +  Vx), 


Bly  ?/  =  log  (2  X  +  V4  .^2  _  1), 

22.  y  =  ]og^^^^, 

V'.T  +  1  +  l 

23.  2/  =  x-[(log.'ir)-'-2  1oga.-  +  2], 

24.  7/  =  log  ("v^o;  +  a  —  a/.c  -  a), 


f/.?/  _  1 

^•»    2V^T^ 

rf//  2 


26:    2/  =  log^t 


+  e- 


dx 

V4:r-l' 

1 
xyx  + 1 

(logaf. 

'/.V 

{x  +  ay  +  {x-a)^ 

dx 

3(a-- 

-a*)! 

/7>. 

, — 

'(a;  +  3)(x  +  2),     - 

=^'^! 

rf;/ 

0„('^"_^    ") 

f/x 

<-■■-  4- 1  -H  c-=^ 

44  DIFFERENTIAL   CALCULUS 


27.   y^log     ^°g'^  ^ 


1  +  log  a;  (/x      .^;  log  x  (1  +  log  x) 

chj_      30  e''' 


28.  ,^(3e--2.-+-2)V2.-  +  l,       ^=^-^^ 

29.  2/  =  3  log  ( Vo;'  +  o  -3)  +.log  (Vx^^+lJ  + 1), 

fh/  4:X 


f^-t'"      x^-2Vx-+3 


30.    2/  =  log  (a  +  V2  aa;  -  a-)  + 


v« 


Va  +  V2  ic  —  a 


(^•t;     2(.i;  +  V2  (to;  -  cr) 


31     ,      ^^^  a.-^  +  1  +  Va^^  +  3  a;-  +  1^       rf?/  ^  ar  -  1 

32.  y  =  logX.«  +  a),  -f  =  ^ J ^^^^ — 

^         °^  "  dx     loi;x[_x  +  a  x 

The  following  may  be  derived  by  XII.  or  by  differentiating  afte 
taking  the  logarithm  of  each  member  of  the  given  equation. 

33.  y  =  x"^   .  ^  =  nx"-^  (1  +  log  x). 

34.  y^iax^r,  g=  (aa.-^)^  [2 +log(aa-^)]. 

35.  y  =  x''^\  ^  =  o.i-'+ni  +  21oga)). 


dx 

-  =  (log  xYi 1-  log  log  X  ]' 

^Vlogx-  •-     J 


36.  y  =  (\ogxy,  .        -.......,, 

dx     ^   °   ^  yiog 

37.  ^  =  ,;(iog  .r,  ^  =  (n  + 1)  (log  .r)"a;<^-  ^>" 


dx 


38.    2/ 


\x  +  a/  ax'      \x  +  aj  \x-{-a     a       x  +  a  J 


* 


DIFFERKXTIATIUN 

The  method  of  diftereiitiatiu-  after  takiii-  the  h)g;iritlii,i 
expression  may  often  be  applied  with  advantage  to  alge^n. 
tious.     This  is  sometimes  called  loijarUhmic  aiff(n-i',.i :.<ti.,.. 

In  this  way  differentiate  Exs.  21-lj(),  pp.  ;30  37. 

39.    Find  the  slope  of  the  catenary  ?/  =  JJ (e"  +  c' "),  at  x  =  0. 
What  is  the  abscissa  of  the  point  wtiere  the  curve  is  inclined  4')° 
to  the  axis  of  X  ?  ^4^ 4^^^*^^  *^=  "  ^°^-  ^^  +  ^^). 

■^    40.    When  does  log„.T  increase  at  tRe  sani«  rate  as  x'! 

^\»s.    When  .r  =  log,,, <'  =  .4.'M.S. 
AYlien  at  one  third  the  rate  '.'  Ans.    When  x=  l.',W2\). 

Verify  these  results  from  logarithm  tables. 

,        41.    If  the  space  described  by  a  point  is  given  by  .s  =  ae'  +  fn'  '. 
y  show  that  the  acceleration  is  equal  to  the  space  passed  over. 

42.    If   a  point  moves  so  that  in  t  seconds   ,s  =  ]0  log  — —  ft-,  ; 

Hnd  the  velocity  and  acceleration  at  the  end  of  1  second.     At  the  end 
of  16  seconds.     Am.   Velocity  =  —  2  ft.,  and  —  .6  ft.  ])er  sec. 
Acceleration  =  .4,  and  .OlT). 


43.    For  what  values  of  x  is     y  =  log  (x  —  2'f  —  — '- ' 

■  (x  —  2Y 

.  increasing  or  a  decreasing  function  ?  v       -y 

Alts.    Increasing  when  a;>3;  decreasing  when  x<  • 


39.    Formulae    for    Differentiation   of   Trigonometric  Functions.     In 

the  following  foi-muhe  the   angle  v  is  supposed  to  be  expressed  in 

circular  measure. 

d    .  (In 

XTTT      — sinM  =  cosM— -• 
V  dx  dx 

d  ■       dn 

XIV      —  cos u=  —  sin  II  —  • 
•     dx  dx 

XV.     —  tan  u  =  sec-  u 

dx  dx 


46  DIFFERENTIAL   CALCULUS 


(I      .  0    dii 

Yvr      — cot  ^/ =  —  cosec- M 

^^^-     civ  dx 

XVII.     — *ec  u  =  sec  u  tan  w  —  • 
dx  ■  dx 

XVI II.      —  oesec u=  —  cosec u  cot n  —  • 
dx  dx 


,rrv       d  .      d.y, 

XiX.     — -versu=smw-— • 

dx  dx 


40.    Proof  of  XIII.     Let  y  =  sin  u, 
then  y  +  A?/  =  sin  («  +  A  w) ; 

therefore  A^  =  sin  (w  +  Am)  —  sin?^. 

But  from  Trigonometry, 

sin  A  -  sin  i5  =  2  sin  ^  (.1  -  5)  cos  ^  (.4  +  B). 

If  we  substitute  A  —  m  +  A  «,  and  i?  =  u, 

we  have  A?/  =  2  cos  [  ^6  +  -—  ]  sin  -— 

.    Am 
^  sni— - 
TT  Ay  /     ,  A«\         2  A7t 

Hence,  -^  =  cos    m. +  ^^ 

'  Ax  \         2  J    An    Ace 


Now  when  Ax  approaches  zero,  Art  likewise  approaches  zero,  and 
as  Au  is  in  circular  measure, 

.    Au 
sm-^  I 

Lim^„^o~^7P  —  1-     See  Art.  12. 

TT  dv  du 

Hence,  -^  =  cos  a  — - 

dx  dx 


DIFFERENTIATION 


41.  Proof  of  XIV.     This  may  be  derived  by  substituting  in  XIII. 


for  li,  -  —  n 


Then 


-f  cos«  =  sin  »/-^-^V -sin  « '^• 
dx  V     dx  dx 


42.    Proof  of  XV.       Since  tan  n  =  ''^"^  ", 

cos  H 

d     .  .        d 

^  cos  «  —  sill  II  —  sill  »'^cos  a 

by  VI.,  7-  tan  u  = 

^         '         dx  cos-u 


,     dii  ,     .  .,     du        du 

cos-  ii  -z-  +  SHI"  "  — ^  

(J.r  dx  (Ix 


o     dii 

=  sec-  II 

dx 

43.    Proof  of  XVI.     This  may  be  derived  from  XV.  by  substiti, 
ins  -—u  for  u. 


44.  Proof  of  XVII.        Since  sec  u  = , 

cos  u 

d  du 

cos?<       sin  u  — 

d  dx  dx 


by  VL, 


u=- 


dx  cos-  u  cos-  u 

du 

=  sec  u  tan  n 

dx 


45.    Proof  of  XVIII.     This  may  be  derived  from  XVII.  by  su 


stituting  ^ u  for  it. 


^  46.  Proof  of  XIX.     This    is  leadilv   obtained  from  XIV.  by  tl 


^•e^^^^ion  vers  M=l -cos  «. 


48  DIFFERENTIAL   CALCULUS 

-^  EXAMPLES 

1.   t/  =  3  sin  3a;  cos  2x  —  2  cos  3a;  sin  2x,  -^  =  5  cos  3a:  cos  2x. 
^  dx 


2.   y  —  log  cos-  X  +  2x  tan  x  —  x^, 
Z.   y  =  log  (sec  7nx  +  tan  mx), 
4.    y  =\og  (a  sin-  x  -\-b  cos^  x), 


_  ,  //I        \   ,  /I  •  c^?/  sin  9 

5.    ?/  —  cos  «  log  sec  (6  —  a)  +6  sm  a,         -^= 

•:  ^         V          y^  '        (16      cos(^-«) 


2x  tan-  x. 

dy_ 
dx 

m  sec  mx. 

dy_ 

2  (a-b)  tana; 

dx 

a  tan-  a;  +  b 

6.  y={m  —  1)  sec^  +  ^i-— (?>i  +  l)sec'"~^a',— —  (wi.-  —  l)sec'"~^a;tan^x 

7.  2/  =  log  tan  (ax  -  -\ 


dx 

dy 


za  sec  MX. 
dx 

8.   r  =  log  [sec  6  tan  ^  (sec  ^  +  tan  Of^ ,   ^'=  («ec  ^  +  tan  g)^ 

oa  tan  6 

Q.    y  =  cosec™  ax  cosec"  bx, 

-^  =  —cosec"*  ax  cosec"  bx  (ma  cot  ax  +  nb  cot  &a;), 
dx 

10.  w  =  2x-  sin  2x  +  2.f  cos  2x  —  sin  2a;,         -^  =  4;k-  cos  2a'. 

da; 

11.  y  =  2  tan^  a;  sec  a;  +  tan  x  sec  x  —  log  (sec  ;c  +  tan  a;), 

^^  =  o  tan- a;  sec'' a;. 
dx 

.  2    ,   _  sin  X  4-  cos  x  dy  _  _  2  sin  a; 

'    ^~  e^  '  dx~  e' 

13.    2/  =  e'''(sin  2a;  -  5  cos  2x),  ^  =13e*^(sin2a;  -  cos2a;) 


14.   y  =  log 


DIFFERENTIATION  49 

,     °  cos  (x  +  a)'  dx      cos  x  cus  (x  -j-  a)' 

15.  ?/  =  sin^  4.j;  cos*  3x,  -^  =  12  sin-  4j-  cos'  3a;  cos  Tar. 

da; 

1  ^  1      sin  a;  +  vers  a;  dv 

16.  ?/  =  log — ,  -^  =  sec  X. 

sin  X  —  vers  x  dx 

17.  y  =  (sin  2  x)^  ^^'  =  ?/  (log  sin  2x  +  2x  cot  2x). 

dx 

18.  y  =  (tan  x)^'"  "=,  .   -^  =  ?/  (cos  x  log  tan  x  +  sec  x). 

ox 

19.  ?/ =  (sin  xy"" '°' "^j  ^  =  ?/(cotxlogcosx  — tanxlogsinx). 

rfx 


nrt  1.  ,1  / 1  +  Sin  X 

20.    7j  =  tan  X  sec  x  +  log  \  -— ' — -. — , 
'  1  —  sin  X 


21 .    ?/  =  (tan  X  —  3  cot  x)  Vtan  x, 
sini^(^  —  «) 


'^  =  2sec'x. 
rfx 


f///  _  3  sec*x 
^^•^•~2tan^x' 

dy  sin « 


22.   ?/  =  log — z — )  T7  = 2' 

sin-(^  +  «) 


23.   y  =  a  log  (a  sin  x  +  b  cos  .x)  +  bx, 


dy  _     a-  +  b- 
dx     a  tan  x  +  b 


sm  i^x- 
OA  V  4y  dy 

24.  ?/=        ^  ^ 


2.  +  I) 


tanJ-2 

25.   2/  =  log = ' 

2tan-^^-l 


rfx      l4-.sin4x 


dy  _         3 
dx     4  —  "» .sin  ./■ 


50  DIFFERENTIAL   CALCULUS 


ofi        _  a  sin  a;  +  6  versa;  <^.y_         2ab\evsx 

a  sin  X  —  h  vers  x'  dx     (a  sin  x  —  b  vers  xf. 

In  each  of  the  following  pairs  of  equations  derive  by  differentia- 
tion eacli.  of  the  two  equations  from  tlie  otli#r: 

27.  sin  2  ic  =  2  sin  a;  cos  a;, 
cos  2  a;  =  cos-x  —  sin^a;. 

28.  sin2a;= , 

1  +  tan- a; 

o         1  —  tan''' a; 

cos  2  a;  = — . 

1  +  tan-'a; 

29.  sin  3  a;  =  3  sin  a;  —  4  siii'^a,-, 
cos  3  a;  =  4  cos^a;  —  3  cos  a;. 

30.  sin  4  a;  =  4  sin  a;  cos" a;  — 4  cos  a;  sin^a;, 
cos  4  .i-  =  1  —  8  sin-.f  cos"a;. 

31.  sin  (ill  -\-n)x=  sin  inx  cos  nx  +  cos  mx  sin  nx, 
cos  {))t  +  n)  X  =  cos  mx  cos  nx  —  sin  mx  sin  nx. 

32.  If  ^  vary  unifornily,  so  that  one  revohition  is  made  in  tt  sec- 
onds, show  that  the  rates  of  increase  of  sin  ^,  wlien  0=  0°,  30°, 
45°,  60°,  90°,  are  respectively  2,  V3^  V2^  1,  0,  per  second. 

33.  If  6  is  increasing  uniformly,  show  that  the  rates  of  increase 
of  tan  0,  when  9  —  0°,  30°,  45°,  60°,  90°,  are  in  harnionical  progres- 
sion. 

34.  For  what  values  of  6,  less  than  90°,  is  sin  6  +  cos  6  an  increas- 
ing or  a  decreasing  function  ? 

Find  its  rate  of  change  when  6  —  15°.  Ans.      ^_ 

V2  ■ 

35.  The  crank  and  connecting  rod  of  a  steam  engine  are  3  and  10 
feet  respectively,  and  the  crank  revolves  uniformly,  making  two 
revolutions  per  second.     At  what  rate  is  the  piston  moving,  when 


DIFFKREXTIATIOX  51 

t 
the  crank  makes  witli  the  line  of  motion  of  the  piston  0°,  •i.j"    90"? 

i;;r)°,  180°';' 

If  a,h,x,  are  the  three  sides  of  the  triangle,  and  d  the  angle 
opposite  b, 

X  =  a  cos  e  +  V6-  —  a-  ain^e. 

Ans.     0,  32.38,  37.70,  20.t)n,  0,  ft.  per  so.- 

36.  A  crank  OP  revolves  about  0  with  angular  velocity  w,  and  :i 
connecting  rod  PQ  is  hinged  to  it  at  P,  whilst  Q  is  constraiiied  to 
move  in  a  fixed  groove  OX.  Prove  tliat  the  velocity  of  Q  is  w.  OR, 
where  R  is  the  point  in  which  the  line  QP  (produced  if  necessary ) 

meets  a  perpendicular  to  OX.  drawn  through  0. 

47.    Inverse  Trigonometric  Functions.      The  inverse  trigonometric 

functions  are  many-valued  functions ;  that  is,  for  any  given  value  of 
X,  there  is  an  intinite  number  of  values  of  sin~^a',  tan~'.r,  &c. 

For  example,  sin"^-  =  -±2m70T-^±2mr,  where  n  is  any  integer. 

Jo  6 

But  if  the  angle  is  restricted  to  values  not  greater  numerically 
than  a  right  angle,  sin~^  x  will  have  only  one  value  for  a  given  value 

of  .r.    Then  sin-'  -  =  -,  sin" /-  ^^  =  -  ^  .     We  thus  regard  sin''  ar, 

2      6'  ^     17  6 

cosec"'  X,  tan"'  x,  and  cot""^r,  as  taken  between  —  ^  and  ^,  that  is,  in 
the  first  or  fourth  quadrants. 

But  cos-'ic,  sec-'x',  and  vers-^a;,  must  be  taken  lietween  0  and  ». 
that  is,  in  the  first  and  second  q[uadrants,  which  include  all  values  of 
the  cosine,  secant,  and  versine. 

These  restrictions  are  assumed  in  the  following  formuhe  of  differ- 
entiation. 

48.  Formulae  for  Differentiation  of  Inverse  Trigonometric  Functions. 

iU 
XX.      ^sin---    -       ''•'•      - 


clx  Vl  —  "' 

du 
XXL     Acos--  '^' 


dx  VI 


52  DIFFERENTIAL   CALCULUS 


XXII. 


XXIII. 


XXIV. 


XXV. 


XXVI. 


du 

-^tan-'M 
dx 

dx 

-1  +  u^ 

du 

— cot"'tt 
dx 

dx 

~        1  +  u? 

du 

^  sec-',  u 

dx 

dx 

«V«^-1 

du 

dx 

dx 

n^t^  -  1 

cr-'' 

du 

^  vers-' 

dx 

dx 

^2u-u' 

49.    Proof  of  XX.      Let  y  =  sin  '  w, 


therefore 

sm  y  =  u. 

By  XIII., 

dy     du 
''''dx-dTx' 

therefore 

du 
dy_   dx   _ 
dx     cosy 

But 

cosy=  ±  Vl 

If  the  angle  y  is  restricted  to  the  first   and   fourth   quadrants 
(Art.  47),  cos  y  is  positive. 


Hence  cos?/=  VI— m^ 

du 

dy         dx 

and  "r""=  "^T^^^"^^ ' 

dx      VI  — w^ 


DIFFERENTIATION 

50.   Proof  of  XXI 

Let  y  =  cos'm; 

tlierefore 

cos  y  =  u. 

By  XIV., 

smy^  =  ^; 
^  dx     dx 

therefore 

du 
dy  _       dx 

58 


dx  sin  2/ 


But  sin  y  =  ±  V  1  —  cos"-//  =  ±  Vl  —  >r. 

If  the   angle   y  is   restricted  to  the  first  and  second  quadrants 
(Art.  47),  sin  y  is  positive. 

^  Hence 


sin  y  = 

VI 

-A 

dy_ 

du 
dx 

and  

dx         VI -w^ 

51.  Proof  of  XXII.       Let  y  =  tan-^  u ; 
therefore  tan  y  =  u. 

By  XV.,  sec^3  =  ^-;L'; 

dx     dx 

du 


iiei 


efore 


dy  _   dx 

dx  ~  sec-//' 


But  sec-  y  =  1  +  tan-  ?/  =  1  +  m*  ; 


therefore 


du_ 
dy  _    dx 


52.    Proof  of  XXIII.     This  may  be  deriv<l  lik.-  XXII..  or  from 
(■<.t-^«  =  tan-'-. 


\ 


54  DIFFERENTIAL   CALCULUS 

53.  Proof   of   XXIV.     This  may  be   obtained  from  XXI.      Since 

sec~^ «  =  eos~^  -, 
u 

d  fl\  1   du  dii 


d        _i  d        _,  1  dx\u  u-d.v  dx 

—  sec  ^  u  =  — cos  '  -  = ^   ^    = = 

dx  dx  u  I        ^ 


/l  _  i_      "^"'  -  1 


54.    Proof    of   XXV.     This  may   be    obtained   from    XX.      Since 
cosec~^it  =  sin~^-) 

d  fV\  1  du  du 


fl  J  ,  ^  _    dx  \uj  u-  dx  dx 

—  cosec"^  u—  —  sin-^ = = 

dx  dx  ^* 


55.    Proof    of    XXVI.     This   may  be  obtained  from  XXI.     Since 
vers~^  u  =  COS"'  (1  —  ''*)> 

d  /-,        s  du 

—(1—u)  — 

d            1          "          1  /,         N                 dx  dx 
-vers"'  u  =  —COS"'  (1  —  u)  = 


dx                    dx  Vl-(l-'w)'       V2u-M=^ 

EXAMPLES  3- 

1            ^       ,5x-l  dy         ^      2 

l.'W  =  tan^' — - — J  -f-  =  ^^T, — :^ — — r- 

^                     2  dx  ■   5.x-  —  2a;  + 1 

o        ■          _i  2a; %               3 

2.   y  =  sec  ^  — > ~" 


3  dx      .'vV4a;2-9 


^  _i  2a;  -  7  cZ?/ 

','y>3,   w  =  sm  '■ — - — ?  -^ 


1 


3  (?a;      V(a;  -  5)(2  -  a;) 

4.    ?/ =  vers"' (8x- —  8a;*),  ,    = —  • 

dx      VI  — ar 

K  ,      _i  a;  —  a  dy  a 

0.   y  =  tan  '  — 


£c  -}-  a  dx     ic^  +  a^ 


DIFFEltENTIATIOX  55 

— ^  y  =  tan-i  (3  tan  6), 

7.   ?/  =  sec-^  sec-  6, 
-1      2 


y  =  vers" 


0;'  +  1 


f?"  — e" 

10.  y  =  cosec"^  — ~,  

11.  y  =  tan-?^:^  +  cot-?^^:i2^       dy^^ 

o  6x  +  l         (ix 


rf2/_ 

3 

dd 

.5 -4  cos  2d 

dy_ 

2 

dd 

Vsee^'  6  +  1 

dy 
dx 

2 

.c^  +  l" 

dy^ 
dx 

2a 

dy_ 

1 

12.   2/  =  cos-^  Vvers  x, 


13.   2/  =  a  tan-i  '-^  -  6  tan-i  -, 
a  •  h 


14.    y=cot-'^  +  «^ 


15.    y  =  sin" 


6.1;  —  a 


sm  a;  —  cos  x 


V2 


".7 

—  4-  VI  +  sec  x. 

dy_ 
dx 

(or  +  a-)(ar'  +  b-) 

a 

x'  +  ar- 

dx 

=  1. 

16.   y  =  sin->  ^•^  +  ^  r^^  =  _i_  ^/?!H' 

6x-  +  a  dx      J,x  +  a\l— X*' 

^— 17.    y  =  tan-'  (sec  x  +  tan  x),  -^  =  -. 

daj      2 

18.  y=sin-       ^      ,  ^'^= 2_ 

e"  +  e-==  dx  ^-'  +  e-' 

19.  y  =  cot  »(x2-x+l)-cot-»(x-l),    ^•?'  =  -J_. 

das      x^4- 1 


56  DIFFERENTIAL   CALCULUS 

20.   y  =  tan-^ ,  ^  -  5  +  4  sin  2x 


21.   t/  =  cos^-3 2^-^,  --        ^ 


Differentiate  both  members  of  the  identical  equations,  Exs.  23-28. 

23.  2cos-^Y^^  =  ''''^'''^- 

24.  3  vers-^  x  =  vers-^  [x{2x-  3)^]. 


25.  sin-^  X  +  sin-i  a  =  sin"^  (aVl  -  a;^  +  a;  VI  -  a^). 

1  ^     -1  (tn  +  n)x 

26.  tan-' ma)  +  tan-Mix  =  tan     l_^^^2' 


27.    ve.s-'2^=2ta„-J^. 

a;  +  3  \     z 

28    tan-^  ^'^^^^"^^  =  tan"^  f ?tan  x)  -  tan'^  ^ • 
a&(l  +  tanx)  V&         /  « 

a^-2a;  +  5  ,  ,      _i^l^    d^         12x^-20 
2»-   ^  =  2^°^^^rp2^T5+''^    "T^'    dx-x^  +  6.'^  +  25 

^       1    4(x-a;5)  #__!_. 


.     i2(a.-^-a2)2  dy       2Va,-^-a^  . 

3V3a2cc  «^     .W4a--x- 

32.    What  value  must  be  assigned  to  a  so  that  the  curve 
y  =  log,  (x  —  7  a)  +  tan"^  ax, 
may  be  parallel  to  the  axis  of  X  at  the  point  a;  =  1  ? 


Ans.  ^  or  —  \. 


DIFFERENTIATION  67 

33.  A  man  walks  across  the  diameter,  200  feet,  of  a  circular 
courtyard  at  a  uniform  rate  of  5  feet  per  second.  A  lamp  at  one 
extremity  of  a  diameter  perpendicular  to  the  first  casts  his  shadow 
upon  the  circular  wall.  Required  the  velocity  of  the  sliadow  along 
the  wall,  when  he  is  at  the  centre ;  when  20  feet  from  centre  ;  when 
50  feet;  when  75  feet;  when  at  circumference. 

Ans.     10,  9/j,  8,  6|,  5  ft.  per  sec. 

56.  Relations  between  Certain  Derivatives.  It  is  necessary  to  notice 
the  relations  between  certain  derivatives  obtained  by  differentiating 
with  respect  to  different  quantities. 

To  express  —  in  terms  of —     If  ?/  is  a  given  function  of  x,  then  x 
clx  dy 

may  be  regarded  as  a  function  of  y.     From  the  former  relation,  we 

have  -^,  and  from  the  latter,  — .     These  derivatives  are  connected 

dx  dy 

by  a  simple  relation. 

It  is  evident  that  —  =-x— > 

Aa;      Ax 

Ay 

however  small  the  values  of  A.^-  and  A?/.     As  these  quantities  ajv 
proach  zero,  we  have  for  the  limits  of  the  members  of  this  equation, 

^  =  1 (1) 

dx     dx 

dy 

That  is,  the  relation  between  -^^  and  -^  is  the  same  as  if  they  were 
ordinary  fractions. 
For  example,  suppose 

—      '■'' 

Differentiating  with  respect  to  y,  we  have 
dx  a 


(2) 


ihj         (y  + 1)' 

dy_^     0/ +  !)••' 

dx  a  x" 


By(l),  ^  =  -ilL±D:=-^„     by  (2). 


58  DIFFERENTIAL  CALCULUS 

This  is  the  same  result  as  that  obtained  by  solving  (2)  with  refer- 
ence to  y,  giving 

and  differentic^ting  this  with  respect  to  x. 

To  express  ^  in  terms  of  -^  and  —  ;  that  is,  to  find  the  derivative 
dx  dz  dx 

of  a  function  of  a  function.     If  y  is  a  given  function  of  z,  and  z  a 

'  given  function  of  x,  it  follows  that  3/  is  a  function  of  x.    This  relation 

may  often  be  obtained  by  eliminating  z   between  the  two   given 

equations,  but  -^  can  be  found  without  such  elimination. 
dx 

Bv  differentiating  the  two  given  equations,  we  find  —and — ,  and 
"^  o  o  J.  dz         dx 

from  these  derivatives,  —  may  be  ob^-^ined  by  the  relation 
dx 

dy_d]idz_  /ON 

dx     dz  dx 

For  it  is  evident  that  — ^  =  ^  — , 

Ax      A2:  Aaj 

however  small  A.i',  A?/,  and  Aa;.    By  taking  the  limits  of  the  members 
of  this  equation  we  obtain  (3).     That  is,  the  relation  is  the  same  as 
if  the  derivatives  were  ordinary  fractions. 
For  example,  suppose 


y  =  ^,  1 

z  =  a-  —  x^.\ 


(4) 


Differentiating  these   equations,  the  first  with  respect  to  z,  and 
the  second  with  respect  to  x,  we  have 

dz  dx 

By  (3),  ^  =  hz\  -2x)  =  -  10x(a-  -  x")',      by  (4). 


DIFFERENTIATION  r>9 

The  saane  result  might  have  been  obtained  by  eliminating  z  betwet-n 
(4),  giving 

y  =  («■  -  •t'")\ 

and  differentiating  this  with  respect  to  x. 

The  rehition  (1)  may  be   obtained  as  a  si)ecial   case   of   (3)  by 
substituting  y  =  x.     This  gives 


Another  form  of  (3)  is 


which  is  of  frequent  use. 


In  Exs.  1-4,  find  —  and  thence  '-^  by  (1). 
dy  dx 


dx  dz      dx 
dz  dx     dx 

=  1. 

dy 
dx_ 
dz 
dx 

CM 

dz' 

EXAMPLES 

(^) 


''"by  —  k'  _    dx      bh  —  ak      (bx  —  a\- 


2.    a;=VrT^i^,  dy^2Vl  +  sin.v^__-_    . 

dx  cos  y  V2  —  7? 


3.   x  = 


y  cZy^(l+16g.v)-^      f 

l+log?/'  dx  logy  i-y-^ 


4.   a,^fl,iog:^i±iL±:^/,  dy^2^f  +  ax^e-'-e' 

■yja  dx  a  - 

In  Exs.  5-8  find  3^  and  ^,  and  thence  -^  by  (3). 


'      -  3z  2x  dy_      12 


2;s-l'  3x-2'  dx      (x-^1) 


9V-' 


60  DIFFERENTIAL   CALCULUS 


6.    2/  =  log'-^±l,    z  =  e  dy_e^-e- 


z  clx      e^  4-  e"" 

7.  i/  =  e-  +  e%   z^logix-x"),     ^  =  4  a;^-6ar^  +  l. 

dx 

8.  ?/  =  log  ,    2  =  sec  a;  +  tan  a,', 

6^  +  a 

dx     2  ab  -j-  (or  +  b^)  cos  cc 

9.  Differentiate  (x^  +  2)-  with  respect  to  a;l 

Let  y  =  (aj^  ^  2)-,  and  2  =  ar\     It  is  required  to  iind  -^- 

^^  =  4  0.(0.^  +  2),  ^  =  3  0.1 

dx  dx 

By(5)  d^^4a-(.or  +  2)^4(.r^  +  2)^ 

dz  3  X"-  3  a; 

10.    Find  the  derivative  of  — „  +  — ,  with  respect  to  '-  +  -• 


\a^^  x-j 


11.  Find  the  derivative  of  sin  3a;  with  respect  to  sin  x. 

Ans.     3  (4  cos^  a-  —  3) . 

12.  Find  the  derivative  of  tan~^y'^.  with  respect  to  log  (1  +  x). 

1 
Ans.      -— r. 
2  -\/x 

io     -r>-    J   ^i       1     •     i.-         i!  1       a  sin  a;  +  &  COS  .T      .^,  ,    , 

13.  Find  the  derivative  of  log  — -. ; with  respect  to 

a  Sin  x  —  b  cos  x 


.          ab  (a-  tan  x  —  b^  cot  x) 
Ans.     — ^^ : ^ . 


d^  sin-  X  —  6^  cos'  x  ci?  +  b'^ 

14.    Given  a;  =  5  cos  </)  —  cos  5<^,  ?/  =  5  sin</)  —  sin  5^;  find  —  . 

dx 

Ans.      — '=t;ui;;0. 
dx 


CHAPTER  IV 

SUCCESSIVE  DIFFERENTIATION 

57.  Definition.  As  we  have  seen,  the  derivative  is  the  result  of 
differentiating  a  given  function  of  x.  This  derivative  being  generally 
also  a  function  of  a;,  may  be  again  differentiated,  and  we  thus  obtain 
what  is  called  the  second  derivatice;  the  result  of  three  successive 
differentiations  is  the  third  derivative :  and  so  on. 


For  example. 

if                       y  =  x\ 

dx 

—  ^  =  12.^•^ 
dxdx 

A  1^=240;, 
dx  dx  dx 

58.   Notation. 

The  second  derivative  of 

denoted  by  '^^ 
dxr 

That  is, 

.^_d_d^^ 
dxr      dx  dx 

Similarly, 

dhj  _  d   d  dy     _  d  dhf^ 
d:x?~  dxdxdx         dxdx-' 

d*y  _  d    d    d  dy  _  d  dhf 
do^~  dxdx  dxdx     dxda^ 

d"y  _  d  d"-^y, 
dx''     dxdx"-' 

61 

62 

Thus,  if 


^^FEKEXTIAL   CALCULUS 

y  = 

:  x\ 

dy_ 

dx 

=  40^, 

d'y 

dx'~ 

=  12a;=, 

dhj 
da^~ 

=  2ix. 

The  successive  derivatives  are  sometimes  called  the  first,  second, 
third,  . . .  differenticd  coefficients. 

If  the  original  function  of  x  is  denoted  hy  f{x),  its  successive  de- 
rivatives are  often  denote^  by 

f'{x),  f"{^^,  f"'(^),  r(^),  '■■  /"(^•)- 

59.    The  nth  Derivative.    It  is  possible  to  express  the  nth  deriva- 
tive of  some  functions. 
For  example, 

(a)  From  y  =  €"%  we  have 

dx  dx^  dx^ 

(b)  From  y  = =  (ax  +  b)-\  we  have 

^  ^  ax-\-b  ^ 

^  =  (-l)a(ax  +  b)-',   f^,  =  (-1)( - 2)a\ax  +  &)-\ 
dx  dx- 

g  =  (-  1)(-  2)(-  3)a%ax  +  6)"^  =  (-  1)13  a^Cax  +  6)"*, 


dx"  ^  J    L-         \  ^       ^  (^^^  ^  ^^J«+l 


SUCCESSIVE   DIFFERENTIATION  63 


(c)   "From  y  =  sin  ax,  we  have 

-^  =  a  cos  ax  =  a  sin  (  a^c-\-  ■ 
dx 


— ^=  a-  cosf  «x  + -  )  =  a-sinf  ax'-f  —  1, 
dor  V  -7  I  -V 

^_  =  cv'toJax  +  '^\  =  aH\n(ax  +^\ 


— ;i  =  a"sin  (  ax  + 
do;" 


EXAMPLES 

1.  y^2x^-hx'  +  20x'-ox'  +  2x,         g  =  120(ar' -  a;  +  1). 

2.  y  =  (0)2-4)2,  ^^=20(ar^-l)(a^-4)i. 

CLX' 

3.  ^  =  .r""  +  a;-'", 

5.    77  =  x*  log  X,  ,  ;  =  — 

•^  ''    '  dot?      X 


6.    ?/  =  a--log(.r-l), 


d^?/_2(j;'-3a;  +  3) 


1.    y  =  \{x-  2y  +  (x  -  l)e^  g,  =  4  .r(e'  +  e^)- 


8.    .r=(^''-3«-  +  ^-3)e^  —  =  4t'i 


9.    r  =  logsec0,  |V  ^  (.  ^^^4  ^  _  4  sec* fi. 


64  DIFFERENTIAL  CALCULUS 

10.    w  =  e-^ll  sin  2  a- +  2  cos  2  x),  ^  =  125  e"^  sin  2  a;. 

11  ^       1  d'y     24x-(l-ar') 

~    12.    ,  =  tan-^l^^,  d^^2(e---  +  .-^-6). 

-^  2      '  da^  (e^  +  e-")' 

^ X  d?y  __  2{x  —  a)(ar'  +  4  ax-  +  a^) 

13.  2/  =  logV..^  +  a;  +  tan-^^,  -^ (x^  +  d^^ 

14.  y  =  (e"*  +  e~"^)  sin  aO,  —^  -\-Aa*y  =  0. 


15.  ^/^aje^CsinaJ  — cosa;)  +  3e''cosx,         — ^  =  4  a^e"^  cos  a;. 

16.  2/  =  e-"'°%  ^,  +  (tana;-l)=^  =  0. 

dar  dx 


1  -  sin  nx  +  cos  wa;  d'^y  ,  2  dy  ,     ,       ^ 

17.    y=- ~ ,  -^  +  -^  +  n-y  =  0. 

X  dx^     X  dx 


18.   y  =  a^(sin  log  x  +  cos  log  a;),       ^^-^  ~  ^  ^—  +  5  2/  =  0. 


19.   2/  =  a^  ^  =  &"(loga)V^ 

da;" 

d"y      (-l)"-'3"|n-l 
, d"y      (-l)"-il.3.5..-(2w-3) 

21.  .  =  V.+i,  5f-°'         ,.(.+i).-l '■ 

22.  2/  =  sin 5 a; sin  2 a;,     £'-^=lf  3"cosC3a;+^V  7"cos/'7a;+^^l. 


SUCCESSIVE   DIFFERENTIATION  65 

The  following  fractions  should  be  separated  into  partial  fractions 
before  differentiating. 

23.   v^-^~,  rf.,_(-ir|»r       1 j_. 

x'-l  dx"  2       |_(a;-l)"+i      (x  +  iy+^j 

^[(2x-3)"+»      (3.c+2)"+'J 


x'-l' 

3x-4 

2x'  +  Sx- 

2' 

13 

6xr  —  5x  — 

g' 

2x2  +  .T  +  l 

dy 


25.   y=      ,    -" ,        ^'  =  (-1) 

2.r  +  2 


26.   y  =  ^-^:  +  -^  +  -^=l4- 


r-1  2ar'-x-l 

dxr      ^       ^  ^[3(x-iy^'     3(2ar  +  l)*+>J 

(x  +  2)2'  dx"  (a;-f2)"+2 

Vax-lJ'  dx"  (ax-iy+^ 

60.  Leibnitz's  Theorem.  This  is  a  formula  for  the  ?jth  derivative 
of  the  product  of  two  factors  in  terms  of  the  successive  derivatives 
of  those  factors. 

A  special  case  of  Leibnitz's  Theorem,  when  n  =  1,  is  Formula  IV.. 

d  /     ^      du     ,     dv  / 1 . 

dx  dx  dx 

For  convenience  let  us  use  the  following  abridged  notation : 


-1' 

dx- 

-!• 

<r-u 

d'u 

66  DIFFERENTIAL   CALCULUS 


Then  (1)  becomes 


—  (uv)  =  UiV-{-uVi (2) 


Differentiating  (2), 

—  (uv)  —  U2V  +  tOiVi  +  Iti^i  +  UV2  =  U2V  +  2  liiVi  +  XIV2, 

d^ 

—  (tiv)  =  u^v  + 1(2^1  +  2  UoVi  +  2u-^Vo  +  U1V2  +  UV3 

=  u^v  +  3  tioVi  +  3uiV2  +  uv^. 

We  shall  find  that  this  law  of  the  terms  applies,  however  far  we 
continue  the  differentiation,  the  coefficients  being  those  of  the  Bino- 
mial Theorem ;  so  that 

—  (uv)  =  u,,v  +  nu„_iVi  +  -^-— — ^u„_2V2  -\ f-  mt{o„_y  +  xiv^.      (3) 

■  This  may  be  proved  by  induction,  by  showing  that,  if  true   for 

—  (tiv),  it  is  also  true  for (uv).     This  exercise  is  left  for  the 

student. 

In  the  ordinary  notation  (3)  becomes 

^^"  fa^.wC^"^\.  I  ^d'^-'ndv      n(n-T)d--'-ud'v 
dx"^    '     die"  dx'^-Ulx  [2       da;"-2diK2 

,     du  d^'^v  ,     d"v 
dx  dx"~^        die" 


EXAMPLES 

1.    Given  y  —  a^sm2x;  find  by  Leibnitz's  Theorem  —^' 

dx* 

d* 
From  (3),  —  (iiv)  =  W4V  +  4  u^v^  +  6  U2V2  +  4  u^Va  +  uv^. 

u  =  x^,  ?<j  =  3  a^,  U2  —  &x,  tig  =  6,  M4  =  0. 
-y  =  sin  2  x,  Vj  =  2  cos  2x,  t;2  =  —  4  sin  2  x,  -Wg  =  —  8  cos  2  aj, 
■V4  =  16  sin  2  a;. 


SUCCESSn'E   DIFFEUENTIATION  67 

£l  =  £i(^-'sin2.r)  =  0.sin2.x-  +  4.r,.2cos2x-  +  (;.Ga-(-4sin2x) 
+  4  .  3  ar  (-  8  cos2x)  +  jr''16  sin  2  x 

=  16  [(.ir'  -  9  X)  sin  2  a;  +  (3  -  6  ar)  cos  2  x] . 

2.    Given  tz  —  xe"^:  find   —  • 

Here         w  =  e"^,     ?^,  =  oe",     ...       ?<„_i  =  a"-'e*",     (/„  =  a"e". 
,      ^  =  3;,       Vi  =  l,       ^2  =  0,      t'3  =  0,     .... 

Substituting  in  (3),  we  have 

dx"      dx"  \       '     J 

I      3.  y^ix^ir^^^,     ^^3(5.>-^-14..  +  13), 


4.   11=6^  log  a;, 


^  =  t'Ao<'  X  +  -  -  -  +  -  -  -^ 


d^n 

6.  y  =  sin  a;  log  cos  a;,         — ^  =  sin  a;  [log  cos  x  —  2  tan-  .»'(3  tan-  r  +  i>)]. 

da;"' 

7.  2/  =  a^a',  ^  =  tt'Oog  a)'-^  [(a;  log  a  +  n)-  -  n]. 

dx" 

'^     {x  +  lf  dx"     ^       '  L      (x  +  l)"^^ 


CHAPTER  V 
DIFFERENTIALS.    INFINITESIMALS 

61.  The  derivative  —  has  been  defined,  not  as  a  fraction  having  a 
dx 
numerator  and  denominator,  but  as  a  single  symbol  representing  the 

limiting  value  of  ^,  as  Ax  approaches  zero.     In  other  words,  the 

derivative  has  not  been  defined  as  a  ratio,  but  as  the  limit  of  a  ratio. 

We  have  seen  (Art.  56)  that  derivatives  have  certain  properties 

of  fractions,  and  there  are  some  advantages  in  treating  them  as  such, 

thus  regarding  -^  as  the  ratio  between  dy  and  dx. 
dx 
Various  definitions  have  been  given  for  dx  and  dy,  but  however 
defined,  they  are  called  differentials  of  x  and  y  respectively.     The 
symbol  d  before  any  quantity  is  read  "  differential  of." 

/  62.  Definition  of  Differential.  One  definition  is  the  following: 
The  differential  of  any  variable  quantity  is  an  infinitely  small  in- 
crement in  that  quantity.  That  is,  dx  is  an  infinitely  small  A.r,  and 
dy  an  infinitely  small  Ay. 

By  the  direct  process  (Art.  16)  of  finding  the  derivative  of  an 
algebraic  function,  Ay  is  generally  expressed  in  a  series  of  ascending 
powers  of  Ax,  beginning  with  the  first. 

Eor  example,  if  y  =  x'^,  y  -{-  Ay  =  (x-\-  Axy, 

and  Ay  =  3x^Ax-{-3x(Axy-\-{Axy.        .      .      .     (1) 

In  finding  the  derivative  we  have 

^  =  Sx''+3xAx  +  (Axy, 
Ax 

in  which,  as  Ax  approaches  zero,  the  second  member  approaches  3x^ 
as  its  limit,  the  second  and  third  terms  approaching  the  limit  zero. 


DIFFEKKNTIALS  09 

If  we  let  Ax  approach  zero  in  equatiou  (1),  i-vfi  v  it-rm  approaches 
zero,  but  there  is  nevertheless  a  marked  distinction  between  them,  in 
that  the  second  and  third  terms,  containing  powers  of  Ax-  higher  tliaii 
tire  lirst,  diminish  more  rapidly  than  that  term.  , 

Thus    we  have  A.>/=3.rA.i-  approximately, 

and  the  closeness  of  the  approximation  increases  as  A.c  approach, 
zero. 

From  this  point  of  view,  regarding  dx  and  dy  as  intinitely  small 
increments,  we  may  write 

d)/  =  3x-dx, 

not  in  the  sense  that  both  sides  ultimately  vanish,  but  in  the  sense 
that  the  ratio  of  the  two  sides  approaches  luiity. 

Thus  .  dy  =  o.rr  dx,    and    —  =  3x-, 

dx 

are  two  modes  of  expressing  the  same  relation. 

According  to  the  first. 

An  infinitely  small  increment  of  y  is  3.r  times  the  cor  respond  in[f  infi- 
nitely small  increment  of  x. 

According  to  the  second, 

TJie  limit  of  the  ratio  of  the  increment  ofy  to  that  of  x,  as  the  latttr 
increment  approaches  zero,  is  3.^. 

Just  as  we  sometimes  say 

"An  infinitely  small  arc  is  equal  to  its  chord,"  instead  of 
"The  limit  of  the  ratio  between  an  arc  and  its  chord,  as  tlio 
quantities  approach  zero,  is  unity." 

So  in  general,  if  y  =/(^)> 

Lim^,=„|^=./-'(.r), 

that  is,  -^=/'(.r)  +  c, 

where  e  approaches  zero  as  Ax  approaches  zero. 


70 


DIFFERENTIAL   CALCULUS 


Hence 


^y  =f'  (a-)  A.T  +  e  Ax, 


and  as  tlie  term  eAx  diminishes  more  rapidly  than  the  term  f'(x)^x, 
we  have 

Ay  =  /  '(a;)  A.e  approxim ately, 

or  dy=f'{x)dx. 

Corresponding  to  every  equation  involving  differentials,  there  is 
another  equation  involving  derivatives  expressing  the  same  relation, 
and  the  former  may  be  used  as  a  convenient  substitute  for  the  more 
rigorous  statement  of  the  latter. 

Thus  the  use  of  differentials  is  not  indispensable,  but  convenient. 
It  should  always  be  kept  in  mind  that^  their  ratio  only  is  important, 
the  derivative  being  the  real  subject  of  mathematical  reasoning. 

63.  Another  Definition  of  Differentials.  The  differentials  fly,  fir, 
are  sometimes  defined  as  any  two  quantities  whose  ratio  equals  the 


derivative 


dx 


Let  us  see  what  this  defini- 
tion means  geometrically. 

If  we  regard  the  derivative 
as  the  slope  of  a  curve, 

^^  =^  tan  RPT. 

dx 
By  this  definition  of  differen- 
tials, dx  may  be  any  distance 
PR  taken  as  the  increment  of 
X,  and  dy  is  then  RT,  the  corre- 
sponding increment  of  the  ordi- 
nate of  the  tangent  line  at  P. 

That  the  two  definitions  are 
consistent  will  appear,  if  we 
diminished. 


suppose    PR    to    be    indefinitely 


7?  7' 

The  smaller  we  take  PR,  the  more  nearly  is  ■ — -  equal  to  unity,  or 

R^ 
in  other  words,  the  more  nearly  is  RT  equal  to  RQ. 

If  PR  is  supposed  to  be  infinitely  small,  this  definition  of  differ- 
entials becomes  that  of  the  preceding  article. 


DII-KKHKNTIALS 


The  second  may  be  said  to  bo  tlie  more  litjorous  of  the  two  dcliu; 
tious,  but  the  first  has  the  advantage  of  being  more  symmetrical,  and 
better  adapted  to  the  various  applications  of  the  calculus  to  mechanic- 
and  physics. 

64.  Formulae  for  Differentials.  The  formuhe  for  differentiation 
may  be  expressed  in  the  form  of  differentials  by  omitting  (/.<•  in  each 
meiubtu'. 

To  each  of  the  formulse  for  a  derivative,  corresponds  a  formula 
for  a  differential. 

Thus  we  have 


II. 
III. 
IV. 

VI. 

VII. 

IX. 


dc  =  0. 

d(n  +  v)  —  du  +  dv. 
d(iiv)  =■  rdii  +  ndv. 
vdn  —  udv 

V' 

iii("~^  du. 


d^H")  = 

d  lo£r  11 


XI.  c?e"  =  e"  du. 

XIII.  d  sin  u  =  cos  u  da. 

XIV.  d  cos  u=  —  sin  ti  du. 
XV.  d  tan  ti  =  sec-  u  du. 

XVI.  d  cot  u  =  —  cosec-  u  du. 

XVII.  d  sec  u  =  sec  u  tan  u  du. 

XVIII.  d  cosec u=—  cosec u  cot u du. 

XX,  fZsin~'«=- 


XXII.     d  tan- 
XXIV.     d  sec-'  u  = 
XXVI.     d  vers-'  u  = 


du 
1  +  u'' 
du 


du 
-s/2u-u' 


72  differp:ntial  calculus 

Differentiation  by  the  new  foinuilce  is  substantially  the  same  as 

by  the  old,  differing  only  in  using  the  symbol  d  instead  of  —  . 

dx 

For  example,  let  y  =  '^     -  . 

"^^-''U  +  sj-  {x^  +  ^y 

^  (.T^  +  3)  c?cc  -  (a;  +  3)  2  xdx 
{x'+'6f 

^  (g;^  +  3  -  2  .r-  -6  x)dx  ^  (3  -  6  a;  -  r;.-^)  dx 
{x'  +  '6f  ~         {x'  +  ^f 

If  we  wish  to  express  the  result  as  the  derivative,  we  have  only 
to  divide  by  dx,  giving 

dy  _^  —  Q>x—x^ 
dx~     (a-2  +  3)='    ■ 

EXAM  PLES 

Differentiate  the  following   functions,  using   differentials  in   the 
process : 

1.   y  =  {x-l){2-^x)(2x  +  ^),    dy=^{-l%x'-{-2x  +  lV)dx. 

o  {t-D{t-2)  6(f-2)dt 

(t  +  l)(t  +  2y  (t+iy(t  +  2y' 

3.   2/  =  Vi^^+T  V^^^=^,  ^y  =  — =^^^-=;^=r dx. 

Vx^  +  1  Vx^  —  2 

cos^^  cos*^ 

5.   2/  =  e'  (a^  -  6  a^  +  24  ic  -  40),     rfy  =  e^  (^l'  +  4 Vza;. 


INFINITESIMALS  7;> 

,6.   r  =  sin  e  log  tan  9,  dr  =  cos  $  log  tau  6  dd  +  sec  6  do. 

7.  y  =  tan-^-i:%,  dy=-l^. 

8.  ?/  =  sin-i  3a;  +  3x'Vl  - 9  ar',      fZ?/ =  GVl -9.r  (/x. 

9.  </,  =  tan-'tan''^;  dd,  = ^  ^an-  $  dO 

tan^e-taii-d-l-l 

^^/^5.  Order  of  Infinitesimals.  In  Art.  G2  we  have  spoken  of  infinitely 
small  or  injin  itesimal  increments. 

An  infinitesimal  may  be  defined  as  a  variable  irhosf  limit  is  zero. 

If  there  are  several  infinitesimals  that  approach  zero  simulta- 
neonslj,  one  of  them,  a,  may  be  taken  as  the  standard  of  comparison 
and  called  the  principal  injinitesivial 

Then  «-,  «^,  «",  are  said  to  be  infinitesimals  of  the  second,  third, 
7iih  orders,  with  respect  to  a. 

In  general  the  order  of  an  infinitesimal  is  defined  as  follows:  An 
pfinitesimal  /?  is  said  to  be  of  the  nth  order  with  respect  to  a  when 

Lim<i=o  -y^  =  Jc,  a  finite  fjuantity,  not  zero.     .     .     •    (1) 

When  n=l,  /?  is  of  the  first  order  with  respect  to  u. 
When  w  =  2,  y8  is  of  the  second  order  with  respect  to  «. 

From  the  definition  it  may  be  shown  that  the  limit  of  tlie  ratio  of 
an  infinitesimal  to  one  of  the  same  order  is  finite,  and  to  one  of  a 
lower  order,  zero. 

Eqnation  (1),  Art.  62,  illustrates  infinitesimals  of  different  orders. 
If  Ave  write  it 

dij  =  3x-  dx  +  3x  (dx)  -  +  (dxy, 

and  regard  dx  as  the  principal  infinitesimal,  the  terms  of  the  second 
member  are  infinitesimals  of  the  first,  second,  and  third  orders,  witli 
respect  to  dx. 

Again,  if  we  regard  x  as  the  principal  infinitesimal,  of  what  orders 
are  sin  a;  and  versa;,  with  respect  to  x? 


74  DIFFERENTIAL   CALCULUS 


By  Art.  12,  Limx=o  ~ —  =  1,  a  finite  quantity. 

Hence  by  (1)  sin  x  is  an  infinitesimal  of  the  firsi  order  with  respect 
to  X. 

T-          vers.r      -r  ■           1  — eosa;  sin-;f      -^  •  1        /sin-^s     1 

Lim  ,_n  — - —  =  Litn     .,  — r-;^ =  Lim 


a;-  *        sin^o;        ar  l  +  cos.i;\   x 

a  finite  quality. 

Hence  by  (1)  vers.r  is  an  infinitesimal  of  the  second  order  with 
respect  to  x. 

Show  that  tan  6  —  sin  6  is  an  infinitesimal  of  the  third  order  with 
respect  to  6. 


CHAPTER  VI 
IMPLICIT  FUNCTIONS 

(See  also  Art.  114.) 

66.   In  the  preceding  chapters  differentiation  has  been  applied  t 
(.\rpUcit  functions  of  a  variable.     The  same  rules  or  forniulie  of  diffci 

entiation  are  sufficient  for  deriving  -^,  ^^,  ~^l,  •••,   when    y  is  an 

dx   dx-   dx' 

implicit  function  of  x;  that  is,  when  the  relation  between  //  and  x  is 

expressed  by  an  equation  containing  these  variables,  but  not  solved 

^vith  respect  to  y. 

For  example,  suppose  the  relation  between  y  and  x  to  be  given  by 

the  equation 

a-y-  +  b'-JT  =  a-b-. 

Differentiating  with  respect  to  x, 

±{cn/  +  b\v^  =  0, 
dx 

2v?y'^-^2b\x  =  0, 
dx 

dy  _      b^x 
dx         ci-y 

Having  thus  obtained  the  first  derivative,  we  may   by  another 
differentiation  find  the  second  derivative. 


-^=-'=-t     "(^-'^ 


dry ^L^ 

dx^~     dxd-y''  «y  t*'/ 

76 


76  DIFFERENTIAL   CALCULUS 

Substituting  now'for  ~  its  value, 

<fy^      b-(aY  +  b-x^^       h* 
dicr  a^if  a?y^-   ' 

By  differentiating  again,  we  may  obtain 

d^y  ^      3  Wx 
dsf  a\f 

The  first  differentiation  may  be  conveniently  performed  by  differ- 
entials instead  of  derivatives.  Thus  we  should  have  from  the 
equation 

a-y-  +  b-x-  —  a-b', 


rmi 


2  a-y  dy  +  2  J)-x  dx  =  0, 


gt¥ing  —  = ^,  as  before. 

dx  a-y 

d-y  d^u 
In  deriving  -^.    --,  ....  derivatives  should  be  used  rather  than 

dr^    di^ 

differentials.    ^^    ^^ 


,  EXAMPLES 

Find  the  following  derivatives. 

1.    (rc^af  +  {y-bf  =  c\ 

dy  _      X—  a     d-y  _  c^  dh/  _     3c^(x  —  a) 

dx         y  —  b'    dx;-  (y  —  by'     dx^  {y  —  by  • 

dx     xy  +  X- 

3.    (cos^)*=(sin<A)^  g^^logsin.^4-<^tang^ 

^        ^        V       ^^  '  ^iQ      log  COS  ^  -  ^  cot  <^ 

^yx4.    ax^+2  hxy+by^=  1, 

dy  _      ax  -f-  Jiy      d-y  _    Jr  —  ah        drx  _     7i^—ab 
dx  hx-\-by'     dxi-      (Jix-\-byy'    dy^      {ax  +  hyy 


IMPLICIT    FUNCTIONS  77 

5.    aor  +  2 /«7/ +  %•- =  0,  '1-(  =  ^1. 

dx     X 

'-  //g.    tan  e  tan  <^  =  m,  '^  =  -  ^"'  ^<t>  'J^^  2  sin  2  </,(co3  2 «»  +  co8  2g) 
(/^  sin  2  ^ '  (/e'  sin"''  L'  (9       • 

;■  /^7.    y -  2..  =  (X - ,/)  log  (.r -  V),     ^'^  =  ^^L=L^    (F.v ^  _ (x-.v)^ 

(/x  y         djT  y^      ' 


9.   7-siu-^  +  2r  +  l  =  0,  f~\'+2rcot6~  =  i^. 


dx        y  4-  2 
7 


;0.    e^*  =  a'6", 


dy  ^  _  ?/  —  log  g     (Py  _  2(y  —  log  n) 
dx  X  —  log  6'   dx^      (x  —  log  b)\ 


11.    log(r«^+r)  =  2tan-^,     <k  =  ^±?l,    ^  =  ^1^+11, 
x      dx     X  —y      dx-       {x  —  yf 


CHAPTER   VII 
SERIES.     POWER   SERIES 

67.    Convergent  and  Divergent  Series.     The  series 

«l+«2+'"3+      •••      +«'« +««+!, (1) 

composed  of  an  indefinite  number  of  terms  following  each  other 
according  to  some  law,  is  said  to  be  convergent  when  the  sum  of  the 
terms  approaches  a  Unite  limit,  as  the  number  of  terms  is  indefi- 
nitely increased.  But  when  this  sum  does  not  approach  a  finite 
limit,  the  series  is  divergent.  That  is,  if  /S„  denote  the  sum  of  the 
nrst  n  terms  of  (1),  the  series  is  convergent,  when 

Lim,,^^  *S',^  =  some  definite  finite  quantity. 

When  this  condition  is  not  satisfied,  the  series  is  divergent. 

Thus  the  geometrical- series, 

a-{- ar -\-ar -\- ai-^  +  ••• 

is  convergent  when  r  is  numerically  less  than  unity,  and  divergent 
when  r  is  numerically  greater  than  unity. 

.„_i_a(l-r") 


1-r 


For  Sn  =  a  +  ar  +  ar-  +  •  •  •  +  a; 

JL  —  ?• 

When  *|r|  <  1,  Um^^S,         ^' 

When  I  r  I  >  1,  Lim„=„  S,,  =  co. 

When  |r|  =1,  the  series  is  also  divergent. 

68.  Series  of  Positive  and  Negative  Terms.  Absolute  and  Conditional 
Convergence.  In  the  case  of  series  composed  of  both  positive  and 
negative  terms,  a  distinction  is  made  between  absolute  convergence 
'^.ud  conditioned  convergence. 

*\r\  denotes  the  numerical  value  of  r, 

78 


SKIIIKS  7<l 

Before  defining  these  terms,  the  following  theorem  should  !>•' 
noticed : 

A  sei'ies  ichose  terms  have  different  siyns  is  conrercjent  if  the  .seru.s 
formed  by  taking  the  absolute  values  of  the  terms  of  the  given  series  is 
convergent. 

AYithout  giving  a  rigorous  proof  of  the  theorem,  we  may  regain 
tlie  given  series  as  the  difference  between  two  series  formed  of  tin; 
positive  and  negative  terras  respectively. 

The  theorem  is  then  equivalent  to  this  : 

If  the  sum  of  two  series  is  convergent,  their  difference  is  also 
convergent. 

A  series  is  said  to  be  absolntehj  convergent,  when  the  series  of  tli 
absolute  values  of  its  terms  is  convergent. 

A  series  whose  terms  have  different  signs  may  be  convergent 
■\Vithout  being  absolutely  convergent.  Such  a  series  is  said  to  be 
conditionally  converged. 

For  example  :  1  — -  +  -—-  + (1; 

U     o     4 

converges  to  the  limit  log^  2, 

but  it  is  not  absolutely  convergent,  since 

is  divergent  (see  Art.  70). 

Series  (1)  is  accordingly  conditionally  convergent. 

But  l-4  +  i-i+- 

2-.     o-     4- 

is  absolutely  convergent  (see  Art.  70). 

69-  Tests  for  Convergence.  Tlie  following  are  some  of  the  mo-' 
useful  tests. 

In  every  conrergeid  series  the  nth   term   must  <ii>iir»<ich   zero  as 
limit,  as  n  is  indefinitely  ini-rcuscd. 

That  is,  the  series  n^  +  n.,  +  «3  H +  ",.  H 

is  convergent,  only  when  Lim,,^^  ?/„  =  0. 

For  S„  =  .V„_,  +  »„. 


80  DIFFERENTIAL   CALCULUS 

If  the  sum  of  the  series  has  a  definite  limit, 

Lim„  ^  ^  >S„  =  Lim„  =  ^  S„,  j. 
Hence  Lim„^^r(„  =  0 (1) 

For  a  (lea-easing  series  whose  terms  are  alternately  positive  and 
negative,  this  condition  is  sufficient  * 

For  example,  1  —  -  +  -  — -  ••• 

is  convergent.     But  the  decreasing  series 

2  _  3      4  _  5 
1      2      3      4"* 

is  divergent,  as  it  does  not  satisfy  (1),  since         Lim„^„  ?<„  =  1. 

The  sum  of  this  series  oscillates  between  two  limits,  loge2  and 
1  -\-  loge  2,  according  as  the  number  of  terms  is  even  or  odd.  Such  a 
series  is  called  an  oscillating  series. 

For  a  series  whose  terms  have  the  same  sign,  the  condition  (1)  is 
not  sufficient.     For  example,  the  harmonic  series 

1  +  -  +  -+-  +  -- 
2      3      4 

is  divergent  (see  Art.  70). 

70.  Comparison  Test.  We  may  often  determine  whether  a  given 
series  of  positive  terms  is  convergent  or  divergent,  by  comparing  its 
terms  with  those  of  another  series  known  to  be  convergent  or  diver- 
gent. 

In  this  way  the  harmonic  series 

i+5+l+i+i+^7+§+- « 

may  be  shown  to  be  divergent,  by  comparing  it  with 

1  +  1  +  1+1  +  1  +  1  +  1+1  +  ... (2) 

^2^4^4^8^8      8^8^  ^^ 

*  The  proof  of  this  is  omitted. 


SERIES  81 

Each  term  of  (1)  is  equal  to,  or  greater  than,  tlie  correspoiulinp; 
term  of  (2).  Hence  if  (2)  is  divergent,  (1)  is  also  divergent.  But 
(2)  may  be  written 

i  +  l+2  +  4      «_       .. 

=1+1+1+1+1+... 

The  sum  of  this  series  is  unlimited;  hence  (2)  is  divergent,  and 
therefore^i). 

Consider  now  the  more  general  series 

i^  +  l  +  ^  +  ^  +  - (^'^> 

If  p  =  1,  the  series  (3)  becomes  (1),  which  is  divergent. 
If  J9<1,  every  term  of  (3)  after  the  first  is  greater  than  the  cor- 
responding term  of  (1).     Hence  (3)  is  divergent  in  this  case  also. 
If  ^  >  1,  compare 

1,1,1,1,1,1,1,1,         ,1,  ,, 

r^  +  2-^  +  3^^  +  r^  +  5-^  +  6^'  +  7T>  +  8-'  +  -  +  157'+-    '    ^'^ 

-ith   l,  +  |,  +  |,,  +  |,  +  L+l_+l^^  +  l-+...+l^^  +  ....    .    (5) 

Every  term  of  (4)  is  equal  to,  or  less  than,  the  corresponding 
term  of  (5).     But  (o)  may  be  written 


j,  +  |;  +  ^„  +  ^„  +  - 


4P 

2 

a  geometrical  series  whose  ratio,  ^,  is  less  than  unity. 

Hence  by  Art.  67,  (5)  is  convergent  and  consequently  (4). 
Thus  it  has  been  shown  that 

when  p  ^1,  the  series  (3)  is  divergent; 

when  p>l,  the  series  (3)  is  convergent. 

The  series  (3)  together  with  the  geometrical  series  are  standard 
series,  with  which  others  may  often  be  compared. 


82  DIFFERENTIAL   CALCULUS 

71.    Cauchy's  Ratio  Test.     This  depcmds  upon  the  ratio  of  any 
term  to  the  preceding  term.     In  the  series 

«l  +  W2+j^+-"+W„  +  W„+i+.--       ....         (1) 
,    .  .       .  ■?<„+! 

this  ratio  is  

Let  us  first  consider,  from  this  point  of  view,  the  geometrical 
sei^ies  a  +  ar  +  m''+-+ar-  +  ar^+'+- (2) 

Here  the  ratio  -—  =  r,  and  is  the  same  for  any  two  adjacent  terms. 

AVe  have  seen  (Art.  67 )  that  tliis  series  is  convergent  or  divergent, 

according  as  \    \  ^  -,        ^    \^   ^ 

|r|<l,  or  |r[>  1. 

That  is,  (2)  is  convergent  or  divergent  according  as 


If  now  (1)  is  any  series  other  than  the  geometrical  series,  the 
ratio  -^  is  not  constant,  but  a  function  of  n.  The  series  is  then 
convergent  or  divergent,  according  as 

Lim„^  I  ^^  I  <  1,  or  Lim„_,  I  "^^  1  >  1.     .  •  .     .        (3) 

We  will  first  suppose  (1)  to  be  a  series  of  positive  terms. 

Let  Lim,,^.^-^^  =  p. 

Suppose  p  <  1.  By  taking  n  sufficiently  large  'we  can  make 
-^  approach  its  limit  p  as  nearly  as  we  please. 

There   must   be  some  value   m,'  of  n,  such  that  when   n  ^  m, 

-—  <  r,  a  proper  fraction. 

Hence  ii„+i  <  n^r,    n„+o  <  u^+iv  <  u^r%  etc. 

«m  +  ^<m+l  +  ^<m+2  H <  ^^  +  U„,r  +  U^  +••••.        •        •      (4) 


SEKIKS  j^3 

But  since  r<l,  the  second  member  of  (4),  which  is  tlie  geometrical 
series,  is  convergent,  and  therefore  tlie  first  member 

is  convergent.     Consequently  (1)  is  convergent. 
Suppose  p>  1.     By  similar  reasoning,  when  ?j  ^  w, 

■*'«  +  ! 

—   >  r,  an  improper  fraction. 
Hence  i/^+i  >  ?<„,  r,    u„^o  >  «„+ir  >  M,„}*^-f-  etc. 

Since  r  >  1,  the  second  member,  and  therefore  the  first  member, 
m^st  be  divergent. 

Thus  the  theorem  is  proved  for  a  series  of  positive  terms. 

If  the  terms  of  (1)  have  different  signs,  it  is  evident  from  Art.  08 
that  the  series  will  be  absolutely  convergent  if 

Lira_|'^|<l. 

It  is  also  true  that  for  different  signs,  (1)  will  be  divergent  if 

Lim„=,  >  1, 

The  proof  of  this  latter  statement  is  omitted. 

If  Lim„_|'^^'i=l, 

the  series  may  be  either  convergent  or  divergent.     There  are  other 
tests  for  such  cases,  but  they  will  not  be  considered  here. 

»    /  EXAMPLES 


1.   Is  the  following  series  convergent? 

JL  +  J_  +  _ 

1-2      2 • 2      3 
Applying  (3),  Art.  71,  we  have 


1  •  2      2  •  2'     3  •  2''  n  2" 


..„       2(7^  +  1) 
Lira, 


n  1 


'2{n  +  l)     2 

As  this  is  less  than  unity,  the  given  series  is  convergent. 
Its  limit  is  log,  2,  as  will  appear  later. 


84  DIFFERENTIAL  CALCULUS 

Determine  which  of  the  following  series  are  convergent,  and  which 
divergent. 

2     I4.  2  ,    3  ,    4  .   ....   f^  By(3),  Art.  71. 

3.  1  ,  ^+li-+li-+....  By  (3),  Art.  71. 

4.  1  +  1  +  1+1  +  ....  By  (1),  Art.  70. 

^3     5     7 

^-  i:2+2:3  +  3:4  +  4:5"^"*' 

a     1^2,3 ,4 ,5         , 
/     6.    l  +  -+2,  +  2-3  +  24+--- 

02         "3         4.4 

8  l_2  +  3_4      5^ By  (1),  Art.  69. 

**•    3     5^7      9^11  ^'  ^ 

9  1  +  1_|-  —  ^ 1-  -^ 1 .  Compare  with  (3),  Art.  70. 

2      5  '  10  n-  + 1 

10     1 1 1— ^  -1 z.  -{ .    Compare  with  (3),  Art.  70. 

■    1  +  Vi      1  +  V2      1  +  V3 

11.    log  2  _  log  I  +  log  I  -  log  ^  +  •  • ..  By  (1),  Art.  69, 


/       12.    sec^-sec^  +  sec--sec-;  + 
/  3  4  5  6 


13.    sin2 1  +  sii^2  ^  +  giiv^  E  +  sin^  |  + 


POWER   SERIES  •  85 

14  1  +  1       2+1   I   3  +  1       4+1 

1^  +  1      22  +  l'^3''^+l      4^  +  l"^'**- 

15  1  +  1    I    2  +  1       3  +  1       4  +  1  ■ 

■    r  +  l"^22+l"^32  +  l"^4'  +  l"*""* 

16.    2+l+l±l_^£+l      5+1 

Answer's 
Exs.  2,  5,  6,  9,  11,  13,  14,  16,  convergent. 
Exs.  3,  4,  7,  8,  10,  12,  15,  divergent. 

Exs.  8,  12,  oscillating. 

72.  Power  Series.  A  series  of  terms  containing  the  positive  in- 
tegral powers  of  a  variable  x,  arranged  in  ascending  order,  as 

tto  +  cijX  +  a.^  +  agar'  +  •  •  •, 

is  called   a  poicer  series  in   x.     The   quantities    «.,   «„  Oj,  •••   are 
supposed  to  be  independent  of  x. 

For  example,  1  +  2  a;  +  3  .^•^'  +  4  or'  H , 

i_r+^_  ?/'+•••, 

are  power  series  in  x  and  y  respectively. 

73.  Convergence  of  Power  Series.  A  power  series  is  generally 
convergent  for  certain  values  of  the  variable  and  divergent  for 
others. 

If  we  apply  the  ratio  test,  (3),  Art.  71,  to  the  power  series 

a^+a^x  +  a.p^-{ ha„iK"H , (1) 

we  have  for  the  ratio  between  two  terms 
u„+i  _a„x  _ 

Lim^hi^l  =  Lim^  1^  I  =  ixlLim,^  I -^ 


S6  •  DIFFERENTIAL  CALCULUS 

The  series  (1)  is  convergent  or  divergent  according  as 

|a;|Liin„_^  -^  <  1,    or    |a;|  Lim„=^  -^  >  1 ; 

that  is,  according  as 

|a;|<Lim„^„p^-  ,    or     |a;|>Lim„ 


The  case '    \x\  =  Lim„^«,  -^  ,    requires  further  examination. 


For  example,  consider  the  series 

1 .+  2  .T  +  3  X'  +  4  ;i;3  +  •••  +  «^"-'  +  (n  +  l)x^  +  ■-.  (2) 

Here     ■ = -,  Lim,,^ =  1. 

a„       n  +  l  n  +  1 

Hence  (2)  is  convergent  or  divergent,  according  as 
]a;|  <  1     or     \x\  >1. 

We  may  say  that  (2)  is  convergent  when  —  1  <  a.'  <  1,  and  the  in- 
terval from  —  1  to  + 1  is  called  the  interval  of  convergence. 


EXAMPLES 

Determine   the  values  of  the  variable   for  which   the   following 
series  are  convergent : 

1.  1  +  X-^X-  +  X^+-: 

2.  J^  +  ^  +  -^+— . 
1.22.33.4 

/     3.  .+  |  +  f +  1'  +  .... 

/v»3         /)«5         nf*i 


l'U\vi:i;  si;i;ii:6  yj 


5.    xA —  •  — h 

2     3      2-45      2.4-67 


V--  a.-2      0.'"       of' 


Exs.  1-5,  convergent  when  —  1  <  a;  <  1. 
Exs.  6-8,  convergent  for  all  values  of  x. 


j 


CHAPTER   VIII        ^ 
EXPANSION  OF  FUNCTIONS 

74.  When  by  any  process  a  given  function  of  a  variable  is 
expressed  as  a  power  series  in  that, variable,  the  function  is  said  to 
be  expanded  into  such  series.  -^ 

Thus  by  ordinary  division 

-^  =  l-a;  +  a^-ar'  +  -.. (1) 

1  +  x 

By  the  Binomial  Theorem 

(x  +  ay  =  a*  +  4  a^x-{-  6  a-x-  +  4  aa^  +  x*. 

(1-x)-'  =  1-{-2x  +  3t'  +  4:x'+-" (2) 

The  methods  employed  in  these  expansions  are  applicable  only  to 
functions  of  a  certain  kind.  We  are  now  about  to  consider  a  more 
general  method  of  expansion,  of  which  the  foregoing  are  only  special 
cases. 

It  should  be  noticed  that  when  a  function  is  expanded  into  a 
power  series  of  an  unlimited  number  of  terms,  as  (1)  and  (2),  the 
expansion  is  valid  only  for  values  of  x  that  make  the  series  con- 
vergent. For  such  values,  the  limit,  of  the  sum  of  the  series  is  the 
given  function,  to  which  we  can  approximate  as  closely  as  we  please 
by  taking  a  sufficient  number  of  terms. 

The  general  method  of  expansion  is  known  as  Taylor's  Theorem 
and  as  Madaurin'' s  Theorem. 

These  two  theorems  are  so  connected  that  either  may  be  regarded 
as  involving  the  other.  We  shall  first  consider  Maclaurin's 
Theorem. 


EXPAXSIOX   OF   FUNXriOXS  89 

75.  Maclaurin's  Theorem.  This  is  a  theorem  by  which  a  function 
of  X  may  be  expanded  into  a  power  series  in  j-.  It  way  be 
expressed  as  follows : 

/(^•)  =./(0)  +/'  (0)-^'  +/'  (0)-^  +/"  (0)-';^  -f ..., 

in  which  f(x)  is  the  given  function  to  be  expanded,  and  /'  (x),/'  (.r), 
f"(x),-'-,  its  successive  derivatives. 

/(0),/(0),/"(0),  ••.,  as  the  notation  implies,  denote  the  values  of 
f(x),  f  (x),  f  {x),  • . .,  when  x  =  0. 

76.  Derivation  of  Maclaurin's  Theorem.  If  we  assume  the 
possibility  of  the  expansion  of  /(.f)  into  a  power  series  in  x,  we  may 
determine  the  series  in  the  following  manner : 

Assume 

fix)  =  A  +  Bx  +  Cx^  +  Dx'  +  Ex'+  ■■;      .     .     .     .      (1) 

where  A,  B,  C,  •••  are  supposed  to  be  constant  coefficients. 

Differentiating  successively,  and  using  the  notation  just  defined, 
we  have 

/'(ic)  =  5  +  2ac  +  3Z>.r  +  4£A''4- •••     ....     (2) 

/"(.T)  =  2C  +  2-3Z)x  +  3-4^ar... (3) 

/"'(.r)  =  2.3i)  +  2.3.4^x  +  ... (4) 

/■-(X-)  =  2.3.4^+- (5) 


Now  since  equation  (1),  and  consequently  (2),  (3),  .-•  are  supposed 
true  for  all  values  of  x,  they  will  be  true  when  x  =  0.  Substituting 
zero  for  x  in  these  equations,  we  have 


from  (1), 

/(0)  =  ^1, 

A=f{% 

from   (2), 

/(0)  =  A 

B=f(0), 

from  (3), 

r'(0)  =  2c, 

e=m 

90  DIFFERENTIAL  CALCULUS 

from  (4),  /'"(O)  =  2  .  3  A       D  =  f^^, 

from  (5),  f^ (0)  =  2-3.4^,     E  =  -Q^, 


Substituting  these  values  of  A,  B,  C,  "  -  in  (1),  we  have 

/(^)=/(0)+/'(0)^+/"(0)|+/"'(0)|+.--.     .     -     ^      (6)1 

77.  As  an  example  in  the  application  of  Maclaurin's  Theorem,  let 
it  be  required  to  expand  log  (1  -\-x)  into  a  power  series- in  x. 

f{x)=\o^{l+x),  /(O)  =  logl  =  0. 

f{oi)  =  :^^={l  +  xr\    .  /'(0)  =  1. 

r{x)^-{i+x)-s  /"(o)  =  -i. 

/'"(.T)=2(l+a;)-^  '    :r'(0)=2.        , 

f'' (X)  =  -  ^(1  +  x)-\  r(0)  =  1 13.^ 


Substituting  in  (G),  Art.  76,  we  have 


O  ^,3 


X-      -zxr     13.T*     I4.^•5 
log(l+aO  =  0  +  l-a;-1.2+^--|X-+^--- 

log(l  +  x)  =x- -  +  ---  +  - . 

sv^;  23      45 

78.  If  in  the  applipation  of  Maclaurin's  Theorem  to  a  given 
function,  any  of  the  quantities, /(O), /'(O), /"(O),  •••  are  infinite, 
this  function  does  not  admit  of  expansion  in  the  proposed  power 
series  in  x. 

In  this  case  fix)  or  some  of  its  derivatives  are  discontinuous  for 
x  =  0,  and  the  conditions  for  Maclaurin's  Theorem  are  not  satisfied 
(see  Art.  94). 

The  functions  logic,  cbtcc,  x-t,  illustrate  this  case. 


f^^i/A^i^fr,  i^XPANSION    OF    Fl  NCTIOXS  91 


EXAMPLES 


Expand  the  following  functions  into  power  serii-s  by  .M;u-l:iiuiirs 
Theorem : 

I  /^»-         'j'3  ^,4 

^      1.    e'  ==  1  +  X  +  ^^  +  y^  +  1^  +  . . ..        Convergent  for  all  valu.'s  of  x. 


'l^  2.    sin X  =  a;  —  ^  +  p;  —  p  4 .  Convergent  for  all  values  of  x. 

''"    3.    cosa;  =  1  —  ^  + "'  —  —  +  • .  •  Convergent  for  all  values  of  x. 

[2      [4      [0 

4.  (a  +  a;)"  =  «"  +  ««"  '•^'  +      "  ~ — ■  (i"~-.r 

,  ?i(n-l)(7i-2)    „  ,  .,  ^,  .     ^      .    , 

H — ^ 7^ a  '"'a;'  +  •••.     Convergent  when  \x\  <  a. 

5.  log„(l+a;)=log„eU'-  2+3"^+"' 

Convergent  when  |a.*I<  1. 

6.  log  (l-a;)  =  -a;-^---- .    Convergent  when  |a;|<l. 

.       7.    tan-'a;=aj--  +  ^-^+"..  Convergent  when 'r!  <  1. 

\ "  3      5       7 


Here  fix)  =  tan~^c, 


/'  (a;)  =  — ^  =  1  -  x'+  x^-a;«+  ••., 

^  ^      1  +  a;- 

/"(x)  =  -2a;  +  4ar^-  Gar'+  ••., 


Q      .      ,  ,  1    a:V  1  . 3    ar'     1  •  3  •  5    af  ^ 


Convergent  wlion  Ij"!  <  1- 


92  DIFFERENTIAL   CALCULUS 

Here  f(x)  —  sin~^x, 

/'(^)  =  -=L=  =  (i-aTi 

.     VI  —  x^ 

Expanding  by  the  Binomial  Theorem, 

/'  (x)  =  1  +  €1.1-+  bx*+  cx'^+  ..., 

2'         2-4'^     2.4-6'  *"' 
/"  (x)  =  2  ax  +  4  bx^+  6  cu-^+  •  •  •, 


where  a  =  - ,  &  = ,  c 


9.    sin  (x  +  a)  =  sin  a  +  x  cos  a  —  —sin  a  —  —  cos  a+  •••. 

[2  [3 

Convergent  for  all  values  of  x. 

10.   log{l  +  x  +  x^=x  +  ^-^  +  '^  +  'l--.. 
J  Convergent  when  |  .r  |  <  1. 

^''l^   11.    e"  sin  .T  =  X  +  ar  +  *—  —  ^ .   Convergent  for  all  values  of  a;.. 

12.  e'  cos  a;  =  1  +  .r  —  ^ ^ — I .   Convergent  for  all  values  of  x, 

3       6 

13.  tan  a;  =  a;  +  ^+^H . 

14.  seca;  =  l+|'4-^+-. 

15.  logseca;  =  -  +  — +  —  +  .... 

^  2^12     45 

Defining  the  hyperbolic  sine,  cosine,  and  tangent  by 

sinh X  =  ^-^^ ,    cosh  x  =  ^  "^  ^    ,    tanh x  =  ^  ~^    ,    show  that 

2      '  2      '  e^+e-^'  ., 

16.  sinhx  =  .  +  |  +  |+..-.  i  ,^l 


EXPANSION   OF   FLNCTIONS 


93 


17.    cosha;  =  l+^  +  -^+... 


18.    tanh  .i-  =  x  -~  +  i^± . 

3       15 


19.    Show  by  means  of  the  expansions  of  Exs.  1,  2,  3,  that 
e'  ^~^  =  cos  X  +  V^  1  sin  z, 
Q-x  y/~l  —  (3Qg  x—^—i  sin  X. 
These  are  important  relations. 

/  79.  Huyghens's  Approximate  Length  of  a  Circular  Arc. 

If  s  denote  the  length  of  the  arc  ACB,   a  its  chord,  and  b 
chord  of  half  the  arc,  it  may  be  shown  that 


tin- 


8&-« 
3      ' 


approximately. 


Let  </)  be  the  half  angle  AOC. 
Then  s  =  2  r<^,  and  by  Ex.  2,  p.  lU, 

«  =  2rsin<^  =  2yYc^-||+|'--' 


6  =  2rsi"^-'? 


2      "    1^2      2='[3      2"'^ 
Combining  so  as  to  eliminate  <\i^, 


i  6  —  « 


If  s  is  an  arc  of  30°,  <^  =  ^  -,  and  the  error  <  .-  t^t^' 
12  10— OUO 


If  s  is  an  arc  of  G0°,  0  =  |^  and  the  error  <^qq' 


94  DIFFERENTIAL   CALCULUS 

80.  Computation  by  Series. 

Compute  by  Ex.  1,  p.  91,  Vt  to  5  decimal  places. 

Ans.  1.64872. 

Compute  ^e  to  10  decimal  places.     ^.  Ans.  1.1051709181. 

Compute  by  Ex.  2,  p.  91,  sin  1°  to  8  decimal  places. 

7r  =  3.14159265.  Ans.  0.01745241. 

Compute  to  4  decimal  places  the  cosiue  of  the  angle  whose  arc  is 
..equal  to  the  radius.  Ans.  0.5403. 

81.  Calculation  of  Logarithms.  By  means  of  the  expansion  of 
log  (1  +  x),  Art.  77,  the  Napierian  logarithms  of  numbers  may  be 
computed. 

Let  us  find  the  logarithms  in  the  following  table, 

log  2  =0.6931, 

log  3  =1.0986, 

log  4  =1.3862, 

logs  =1.6094, 

log  6  =1.7917, 

log  7  =1.9459, 

log  8  =2.0793, 

log  9  =2.1972, 
log  10  =  2.3025. 

It  is  only  necessary  to  calculate  directly  the  logarithms  of  the  prime 
numbers  2,  3,  5,  7,  as  the  others  can  be  expressed  in  terms  of  these. 
We  have  from  Art.  77, 

log2  =  log(l  +  l)=l-^  +  ^-^+.... 

This  series  is  convergent,  but  converges  so  slowly  that  100  terms 
would  give  only  two  decimal  places  correctly.  But  we  may  obtain 
a  series  converging  much  more  rapidly  by  taking 

log2=log !■  =  log  (1  +  i)-log  (1  -  i). 


EXPANSION   OF ,  FUXCTK )x\S 


Do 


For  log  ^±1  =  log  (1  +  X)  -  log  (1  -  .T) 


2       •'       ' 


x^      x^      X* 


4    ■  ^  2       3      4  ^ 


3       o 


=  2(ar+|-  +  ?^+-)- 


Couveigent  when  |a;|<l. 

Thus  log2  =  2/^i  +  -^  +  -A-  +  ^-+...\ 

Four  terms  of  this  series  give  log  2  =  .G931. 
The  computation  may  be  arranged  as  follows: 


X 

3"" 

.333333 

i  =.333333 

1' 

3« 

.037037 

,',,-M23i6 

1 
3' 

.004115 

^  =  .000823 
5  •  3' 

1 
3- 

.000457 

^  .  -  .000065 

7  -o' 

1  _ 

3" 

.000051 

-^  =  .000006 
9 . 3"-' 

.34657 

2 

.69314 


The  numbers  in  the  first  column  may  be  obtained  by  dividing  suc- 


cessively by  9. 


1  +.T 


1-f  T7 


Any  number  may  be  put  in  the  form -^  and  log.'*.  = 

may  be  found  like  log  2. 


96  DIFFERENTIAL   CALCULUS 

But  haviug  log  2,  it  is  easier  to  compute 

log -  =  log -, 

and  then  log  3  =  log  -  +  log  2. 

Let  the  student  make  this  computation. 

5  ^  +  1 

Find  log  5  from  log  -  =  log • 

^  1-- 

4 

In  a  similar  way  find  log  7  from  log  5. 

Having  obtained  the  logarithms  of  2,  S,  5,  7,  find  the  other 
rithms  in  the  table  at  the  beginning  of  this  article. 

To  obtain  the  common  logarithm,  that  is,  logarithnijo,  it  is  only- 
necessary  to  multiply  the  Napierian  logarithm  by  .4343,  the  modulus 
of  the  common  system. 

Find  thus  the  common  logarithms  of  the  numbers  m  the  foregoing 
tables,  —  first,  of  2,  3,  5,  7,  and  from  these  the  others. 

82.  Computation  of  w.     From  Ex.  7,  p.  91,  by  letting  x  =  1,  we  have 

a  slowly  converging  series. 

To  obtain  a  series  converging  more  rapidly,  we  may  use 
tan-^  1  =  tan-^  -  +  tan  -'  -, 

from  which  l^2~'3^^'^5^'~7^''^"' 

+  1_J_  +  J ^+.... 

^3     3-33^5.35     7-3^ 


EXPANSION   OK   FIJNCTIONS 


97 


By  taking  9  terms  of  the  first  series  and  5  of  the  s.'c-on.l    the 
student  will  find  ' 

^=0.4G3G47...  +0.321751... 
and  7r  =  3.141o9  .... 

Other  forms  of  tau-^  1  may  be  used,  giving  series  converging  even 
moire  rapidly,  as 


tan-il  =  2  tan-'l  +  tan-i^. 


tan-^  1  =  4  tan-^     —  tan- 

By  these  formulae  the  computation  has  been  carried  to  200  deci- 
mal places. 

y  ' 

-'83.  Taylor's  Theorem.  This  is  a  theorem  for  expanding  a  function 
of  the  sum  of  two  quantities  into  a  power  series  in  one  of  these 
quantities. 

As  the  Binomial  Theorem  expands  (x  +  /«)"  into  a  power  series 
in  h,  so  Taylor's  Theorem  expands  f(x  +  Ii)  into  such  a  series.  It 
may  be  expressed  as  follows  : 

h-  h^ 

fix  +  h)=f{x)+f'(x)  h  +f"{x)^  +/"'  (•^)|3  +  -. 

84.  The  proof  of  Tajior's  Theorem  depends  ujwn  the  following 
principle : 

If  we  differentiate  f(x  +  h)  with  respect  to  x,  regarding  h  con- 
stant, the  result  is  the  same  as  if  we  differentiate  it  with  respect 
to  h,  regarding  a;  constant. 

That  is,  4-  f(^  +  '0=4  /(■''  +  '0- 

clx  dh 

For,  let  z  =  x  +  h, (1) 


98  DIFFERENTIAL   CALCULUS 

then  by  (3),  Art.  56, 

But  from  (1),         —  =  1      and     ^  =  1; 
^  ^'         dx        '  dh        ' 

therefore  ~  f(x  +  /i)  =  4  /C^'  +  '0- 

dx  dh 

85.  Derivation  of  Taylor's  Theorem.  If  we  ass\ime  the  possibility 
of  the  expansion. of  fix  +h)  into  a  power  series  in  It,  we  may  deter- 
mine the  series  by  the  aid  of  the  preceding  article.     Assume 

f(x^h)=A  +  Bh+Ch-  +  Dlv'+  -. (1) 

where  A,  B,  C,  •••  are  supposed  to  be  functions  of  x  but  not  of  h. 
Differentiating  (1),  firgt  with  respect  to  x,  then  with  respect  to  h, 

d    .f     ,  ,,      dA  ,  dB.   ,  dC-,0  ,  dD.n  , 
dx  dx      dx        dx  dx 

By  Art.  84,  the  first  members  of  these  two  equations  are  equal  to 
e^h  other,  therefore 

^  +  ^/i  +  ^7^2  +  ...  =  5  +  2  C/i  +  3Z>7i2  +  .... 
dx       dx  dx 

Equating  the  coefl&cients  of  like  powers  of  h  according  to  the 
principle  of  Undetermined  Coefficients,  ^v€Jlave 

—  =B  B=—- 

dx  '  dx' 

^  =  20  C=-  — 

dx  '  2dx'' 

dx  '  [3  dx^ 


EXPANSION    OF    KL'NCTIONS  99 

The  coefficient  A  may  be  found  from  (1)  by  putting  h  =  0,  as  that 
equation  is  supposed  true  for  all  values  of  /*. 

Then  .      A  =f(x). 

Hence  23  =  —  =  f>(x\ 

dx      -^  ^  ^ 


Substituting  these  expressions  for  A,  B,  C,  •••  in  (1),  we  have 

86.  Maclaurin's  Theorem  may  be  obtained  from  Taylor's  Theorem 
by  substituting  a;  =  0.     We  then  have 

f(h)  =/(0)  +r(o)h  +/"(o)|  +f"'m~  +  -' 

This  is  Maclaurin's  Theorem  expressed  in  terms  of  h  instead  of  .r. 

87.  As  an  example  in  the  ap|)lication  of  Taylor's  Theorem,  It-t  it 
be  required  to  expand  sin  (.»;  +  h)  into  a  power  series  in  h. 

fix  +  h)  =  sin  {X  +  h)  ; 
hence  f{x)  =  sin  .r, 

/'(x)  =  cos  X, 
f"(x)  =  -  sin  X, 
f!"(x)  =  —  cos  X, 
f^{x)  =  sin  X. 


Substituting  these  expressions  in  (2),  Art.  85,  we  find 

sin  {x  +  h)  =  ^mx  +  h  cos  a;  -  |-  sin  x  -  ^  cos  a;  -|-  -  sin  x  +  ••  • 

lii.  '_  lL 


100  DIFFERENTIAL   CALCULUS 

EXAMPLES 
Derive  the  following  expansions  by  Taylor's  Theorem  : 

/r  h^ 

\      1.    cos  (a;  + /i)  =  cos  a;  — /isina;  — -r^cosa;  +  —  sin  a;  +  •••. 

3.  {x -{- hj  =  x' -\- 1  x^  h -\ . 

4.  {x  +  lif  =  X-  +  nx^^-'  h  +  "  ^"  ~-^^ 7r  +  •  •  • . 

0  5.    log(:-c  +  /0  =  loga;  +  ^--'^,  +  „'''-3--^,  +  ---. 
^  '  X      2  x^     3  XT     Ax* 

•  6.    tan  (a;  +/i)  =  tan  a;  +  Jisec^x  +  /rsec'a;tan  a; 

+ 1^3  sec^  a;  -  2  sec2  a;)  +  . . .. 
o 

7.  Compute  from  Ex.  1,  cos  012°  =  0.4695. 

8.  Compute  from  Ex.  6,  tan  44°  =  0.9657,  tan  46°  =  1.0355. 

^  [±  li 


As  a  special  case  of  Ex.  10,  derive 

2        x  —  h      X     3  af     5  xf 
11.  /(2a;)  =  /(a;)  +  a/'(a;)  +  ^  f"  (x)  +  ^  /'"  (a;)  -f-  -. 

af^       /'"(a;) 
(1  +  a;)==      [3 
13.    Uy=  f(x),    show  that 


+ 


^  Aa;  +  ^'  '^^^')"  +  ^(^^ 
dx  dx^    \2_       dx"    [3 


EXPANSION   OF   rUNCTlONB 


101 


88.  In  the  preceding  derivations  of  Taylor's  and  ^klaclaiirin 
Theorems,  the  possibility  of  the  expansion  in  tin-  i)roposo(l  fcmi  li; 
been  assumed.  In  the  remainder  of  this  chapter  we  shall  show  ho 
Taylor's  Theorem  may  be  derived  without  sucli  assumption. 


89.    Rolle's    Theorem.       If  a  given    function   <^(.c)    is   zero 
x  =  a  and  when  x  =  h,  and  is  continuous   between   those  vali 
well  as  its  derivative  ^'  (x) ;  then 
4>'(x)  must  be  zero  for  some  value 
of  X  between  a  and  b. 

Let  the  function  be  represented 
by  the  curve  y  =  <f>  (x).  Let 
OA  =  a,  OB=h.  Then  accord- 
ing to  the  hypothesis,  y  =  0  when 
X  =  a,  and  when  x  =  b. 

Since  the  curve  is  continuous 
between  A  and  B,  there  must  be 

some  point  P  between  them,  where  the  tangent  is  parallel  to 
and  consequently  <i>{x)  =  0. 


when 
•s,  ;i^ 


OX, 


90.    Mean  Value  Theorem.     Ji  fix)   is  continuous  from   x  =  a  to 
X  =  b,  there  must  be  some  value  x^  of  x,  for  which 


This  may  be  stated  geometrically 
thus : 

The  difference  of  the  ordinates  of 
two  points  of  a  continuous  curve, 
divided  by  the  corresponding  differ- 
ence of  abscissas  of  these  points,  equals 
the  slope  of  the  curve  at  some  inter- 
mediate point. 

In  the  figure  let  the  curve  PRQ 
.represent  y=f(x). 


J 

f 

1 

M 

^^ 

K      B 


w 

102         ^  DIFFERENTIAL  CALCULUS 

Let  OA  =a,  0B  =  b.     Then 

b-a  PM  ^ 

At  some  point  of  the  curve,  as  J?,  between  P  and  Q,  a  tangent  can 
be  drawn  parallel  to  PQ.  Call  OK=x^.  Then  the  slope  of  the 
tangent  at  R  \s,f\x^,  which  equals  tan  QPM. 

Hence  -^  ~/(^)^/(.r^),  where  a<x^<b.      .     (1) 

If  we  let  AB  =  b  —  a  =  7i,  b  =  a  +  h,  (1)  may  be  written 

/(«  +  h)  =f(a)  +  hf(a  +  <f>h),         ~  where  "  0  <  c^  <  1.    .    (2) 

91.  Another  Proof.  The  following  method  of  deriving  (2),  kxtlm^l 
is  important,  in  that  it  may  be  extended  to  higher  derivatives  of 
J'{x),  as  appears  in  Arts.  92,  93, 

Let  R  be  defined  by 

f(a  +  h)-f{a)-liR  =  0 (1) 

That  is,  let  R  denote  Aa  +  li)-f{a)^ 

h 

Consider  a  function  of  x  whose  expression  is  the  same  as  (1)  with 
X  substituted  for  h.     Call  this  function  <^(.f). 

That  is,  </)(.i')  =/(a  +  .r) -/(«)- a-i? (2) 

Differentiating,  <k'(^)  —fi'^'-  +  ^)  —  R (3)  ' 

It  is  evident  from  (2)  that  <j>(x)  =  0,  when  a-  =  h,  by  (1)  ; 

also  (f>(x)  —  0,  when  x  =  0. 

Hence  by  Rolle's  Theorem,  Art.  80,  4>'{.v)  =  0,  for  some  value  of  x 
between  0  and  h. 

Calling  this  value  of  x,  Oh,  we  have  from  (3) 

/'(a  +  Oh)  -R  =  0. 

Substituting  this  value  of  R  in  (1), 

f(a  +  h)=f(a)  +  hf'(a  +  Oh). 


EXPANSION   OF   FUNCTIONS  lOU 

92.  Extension  of  Mean  Value  Theorem.  A\'e  may  extend  the  metlKnl 
of  the  preceding  article  so  as  to  include  the  second  derivative,  and 

obtain  /(a  +  /*)  =/  (a)  +  hf  (a)  +^V"  («  +  ^'O- 

Define  7?  by    /(a  +  /i)-/(a)-/</'(a) -^7e  =  0 (1) 

Let        4>{x)=f{a  +  x)-f{a)-ccf(a)-'~R (2) 

Hence    <i>'{x)  =f  (a  +  x)  -/'  (a)  -xR, 

<l>"(x)=f"(a  +  x)-B (3; 

From  (2)  it  is  evident  that  <f)(x)  =  0,  when  x  =  h,     by  (1); 
also  (f,  (x)  =  0,  when  x  =  0. 

Hence  by  Rolle's  Theorem,  Art.  89,  0'(.r)  =  O,  for  some  value,  x„ 
of  X,  between  0  and  h. 

Also      <^'  (x)  =  0,  when  x  =  0. 

Hence  (f>"  (x)  =  0,  for  some  value,  Xi,  between  0  and  a*,,  that  is,  be- 
tween 0  and  h.     Writing  x.,  =  6h,  we  have  from  (3), 

/"  (a  +  Oh)  -R  =  0. 
Substituting. this  value  of  R  in  (1),  we  have 

/(a  +  h)  =J\a)  +  hf  («)  +  ^./""(a  +  Bh). 

It  is  to  be  noticed  that  it  is  assumed  that /(x),/' (a-),  and/"(j-)are 
continuous  from  x  =  atox  =  a  +  h. 

93.  Taylor's  Theorem.  This  may  now  be  derived  by  extending  the 
preceding  method  so  as  to  include  the  ?(th  derivative.  * 

It  is  assumed  that  f{x)  and  its  first  n  derivatives  are  continuous 
from    x—  a    to    x  =  a  +  h. 

Define  R  by 
f{a  +  h)-f{a)-hf'{a)  -^J"{n) -^/""'(a) -^"i?=0.    (1) 


104  DIFFERENTIAL   CALCULUS 

Let 
c^(x)^/(a  +  .T)-/(a)-a:r(a)-|/"(a) |£^/"-Ha)-^y"ig, 

<j.Xx)  =f'(a  +  x)  -/'(«)  _.x/" (a) ^-£^r-^(a)-^f^i?, 

<^"(x-)  =/"  (a  +  X)  -  f"  (a) -^  /"  -1  (a)  -  -^E, 


\n-3- 


n 


r~'  i^)  =/'"'  («  +  ■«)  -/"^'  (a)  -  a.-i?, 

<^"(.^^)=/«(a  +  a;)-J? (2) 

As  in  the  preceding  articles,  it  is  evident  that 

<^(.r)  =  0,  when  x  =  h,  and  also  when  a;  =0, 
Hence   (}>'(x)  =  0,  when  x  —  x^,  where  0  <  x^  <  h. 
But        <i>'(x)  —  0,  when  x=0  ;  hence 

<fi"(x)  —  0,  when  x  —  x^,  where  0  <  o.^  <  a^.    ■ 

Continuing  this  reasoning,  we  find. 

<^"  (x)  =  0,  when  a:  =  .t„,  where  0  <  .t„  <  iK„_i, 
that  is,  where  a-,,  is  between  0  and  h. 
Hence  from  (2)  </>"  {6h)  =/"  (o  +  61i)  -E  =  0. 


Substituting  this  value  of  R  in  (1),  we  have 

J  I  n  —  1  I  n 

Since  a  is  any  quantity,  we  may  write  x  in  place  of  a,  giving 


f{x+h)=f{x)+hr{x)  +|/"(a.)+ ...  +J^r-\x) 

+  ^/"(.,  +  ^/,).    .....     (3) 

\n_ 


EXPANSION   OF    FLXCTIOXS  lof, 

94.    Remainder.     The  last  term  of  this  equation 

\n 

is  called  the  remainder  after  n  terms  in  Tanlors  Theorem.  Wlion  t! 
limit  of  this  remainder  is  zero,  as  n  is  indetinitely  increased,  Tayloi 
Theorem  gives  a  convergent  series. 

We  have  already  seen  (Art.  86)  that  Maclaurin's  Theorem  is  a 
special  case  of  Taylor's  Theorem,  so  that  corresponding  to  (.3)  of  tin- 
preceding  article,  we  may  write  IMaclaurin's  Theorem 

f(x)  =  /(o)  +  .T/'(0) +,^/"(0)  +  -  +  ^f''-\o)+f'f''(ex), 

f{x)  and  its  first  n  derivatives  being  assumed  continuous  for  valup  ■ 
from  0  to  X. 

Thus  the  remainder  after  n  terms  in  Madaurin's  Theorem  is 

i|/-(te). 


When  Lim 

Madaurin's  Theorem  gives  a  convergent  series. 
Applying  (1)  tof{x)  =  e",  we  have 


's  a  convergent  serii 

e",  we  have 

Lim„_^  r-\^»^1  =  0, 
L^      J 


which  is  evidently  satisfied  for  all  values  of  .r. 

The  same  is  true  for   f(x)  =  sin  x,    and   ./■(•'")  =  cos  x. 
If/(.r)  =  log(l  +  aO,  (1)  becomes 

Lim„_.  [f^  (-ly-'llLzll 


=--[^'fe)"]-- 


This  is  satisfied  when  |a?|  <  1. 

It  is  to  be  noticed  that  the  preceding  test  for  convergence  is  of 
practical  use,  unless  the  nth  derivative  of  J'(x)  can  be  expressed. 


CHAPTER  IX 
INDETERMINATE   FORMS 

95.  Value  of  Fraction  as  Limit.    The  value  of  the  fraction  '^^'"'   for 

any  assigned  value  or  x,  as  x  =a,  is  ^     '  • 
_  ,  '/'(«)   . 

This  is  a  definite  quantity,  unless  <^(a)  or,j^.a)  is  zero  or  infinity. 
When  this  is  the  case,-  we  may,  by  regarding  the  fraction  as  a 
continuous  variable,  define  its  value  when  x  equals  a,  as-  its  limit 
when  X  ajyproaches  a. 

That  is,  the  value  of  .^W_,  when  x  =  a,  is  defined  to  be 

ip{x) 

Lim_._„  *P  •-'       or  what  is  the  same  thing, 

Lim,_„^i^^+^. 

There  is  no  difiiculty  in-  determining  this  limit  immediately,  when 
the  numerator  only,  or  the  denominator  only;  is  zero  oi'  infinity  ;  or 
when  one  is  zero  and  the  other  infinity. 

We  will  now  consider  the  cases  where,  for  some  assigned  value  of 
x,  the  numerator  and  denominator  are  both  zero  or  both  infinity. 
The  fraction  is  then  said  to  be  indeterminate. 

96.  Evaluation  of  the  Indeterminate  Form  -  •  Frequently  a  trans- 
formation of  the  given  fraction  will  determine  its  value. 

Thus,  ■ — =  - ,  when  x  =  1. 

'  •  a;--l  0' 

But  if  we  reduce  the  fraction  to  its  lowest  terms,  we  have 


Lim,_, — '^ =  Lim 


x  +  2     3. 


x=l- 


'X-\-l      2 
lOG 


INDETERMINATE    KOUMS  107 


Again,  ,  ^      *" —  =  -,  when  x  =  2. 


^ic  - 1  - 1     0 


By  rationalizing  the  denominator, 

Lun._,      ZZ-—  =  Lin>.  „  0^- 2)  ( V^m  +  1} 

'■y/x-1-1  '  X-2 


=  Lim,=2(Va;-l  +  1)  =  2. 


As  another  illustrati 


on, 


cos  2  5  0       ,       ^     ff 

- ,  when  0  =  -' 


cos  5  —  sin  5      0  4 

i>   .  -r  •  cos  2  5  T  •  cos-  5  -  sin^  6 

•1  cos  6  —  sin  6  *  cos  6  —  sin  6 

=  Linig^ir  (cos  5  +  sin  $)  =  cos- +  sin-=  ^/2, 
i  4  4' 

Tlie  Differential  Calculus  furnishes  the  following  general  method  : 

97.   Form  a  new  fraction,  takinr/  the  derivative  of  the  r/iceii  numerotiu 
for  a  neio  nnmerator,  and  of  the  given  denominator  for  a  new  denoih 
nator.     The  value  of  this  neio  fraction,  for  the  assitpwd  ndue  oi  il> 
variable,  is  the  limitimj  value  of  the  (jiven  fraction. 

AVe  will  now  show  how  this  rule  is  derived. 

Suppose  the  fraction  ^  '' '  =  -^  when  x  =  «;   that  is,  <^(tt)  =0,  and 

By  Art.  95  the  required  value  of  the  fraction  is  the  limit   of 

■  -     '    ^^ ,  as  h  approaches  zero. 
^{a  +  hy  ^^ 

By  the  JNIean  Value  Theorem,  (2),  Art.  90 

<^  (a  +  h)  =  <t>  (a)  +  hcf>'(a  +  dh), 
t{,(a  +  h)  =  ip(a)  +  /<V'(a  +  OJt), 
where  6  and  Oi  are  proper  fractions. 


r; 


108  DIFFERENTIAL   CALCULUS 

But  since  ^(o)  =  0  and  i//(a)  =  0,  we  have 

cjyja  +  h)  ^  h4'(a  +  eh)  ^  cf>'(a  +  Oh)  ^ 
i{f(a  +  h)      lixp'{a  +  dji)      xp'ia  +  eji) 

Hence  Lim^^i^^-±^  =  ^^, 

wliicli  is  the  theorem  expressed  by  the  rule. 

If  (ji'(a)  =  0  and  </''(«)  =  0,  it  follows  likewise  that 

Lim,-o^^^^^+^  =  -^^; 

that  is,  the  process  expressed  by  the  rule  must  be  repeated,  and  as 
often  as  may  be  necessary  to  obtain  a  result  which  is  not  indeter- 
minate. 

For  example,  let  us  find  the  limiting  value  of  the  fraction  in 

Art.  96. 

cb(x)      ^2^.T-2     0      1  ^ 

— --^  —  — —  -,  when  X  =  1. 


^//(.^•)  X!'  —  l         0 

«A'(-«) 

■    3 

Thus  the  required  limiting  value  is  -  • 


^  ^  ^  —  — — ! —  =  -,  when  x  =  l. 


of 


For  another  example,  let  us  find  the  limiting  value,  when  x  =  0, 
e  +  e-"  -  2 


I 


1  —  cos  X 

<f>(x)  _  e^  +  e~"  -  2  _ 
il/(x)        1  —  cos  X 

0 

when  X  =0. 

4>'(x)  _e''  —  e-^  _0 
il/'{x)         sin  a;        O' 

when  x  =  0. 

<l>"(x)  _  6==  +  e--  _  ^ 
il/"(x)        cos  X        "' 

when  X  =  0> 

i 

Thus  the  required  limiting  value  is  2. 


INDKTKUMI.NATE    FORMS  lOJ) 


EXAMPLES 


Find  the  liniiting  values  of  the  following  fractions  for  the  assigned 
values  of  the  variable. 


1.    —5^— — -;      " ,  when  a;  =  2.        vbf.s.     74  +  -. 

XT  —  Ix  2 


x(x-l)"-2  1  o  <  1 

-i — ; ^ — — .  wnen3^  =  z.  Aux      7*4-- 

or  —  2x 

I,  2    -'^g(^-^"  + 


«     log  (3  .X-- +  it- —  •>)  1  1  ,         ^ 

2.    — ^-^^ ! -,  when  .t-  =  1.        ^l/(.s.    7. 


/     ;f^,    ^ — 1,  when  a- =  0.         .I/^n.     log^  a. 

b'  —  1 


.  ^-tan_^^  when  a- =  0.        A 

-^  X  —  sin~^  x 

_^.  g^  log(a;-^-4x  +  5)^ 
'^      '      losr  cos  (x  —  2) 


ar 


sm- 


sm  »i«  — sm  «« 


Avhen  X  =  2.        ^Ihs. 


log  cos  {x  —  2) 
g_    xe^-\og(x  +  l)^  when.r  =  0.         .W.     '1 


y     e  vers  ^  when^  =  0.        .Iws.     0. 


8.    ^"^  ""^  ~ '^^^ '"%  whenm  =  ».        ^l«s.    cos  no. 

sin(?/i— ?()« 

/.  Q     sin  ( ^  +  3  )  - 1  _  ^  1 

log  sin  2^  -4  4 

:j>^^"'  «"_;,«'  i-iog6 


/j:  jj^    log, g- log,,  &^  ^l^gn  a  =  b.        Ans. 


o 


a-b        '  '  ^'log^ 

2x^  —  i  J-  +  9  .r  - 
a;^-2ar*  +  2a;-l 


'al12     -^'^  -  -  .v^-4  .^-'  +  9  ■^•-4    ^y^^^,^  ^.^  1,        ^1„,.     4. 


110  DIFFERENTIAL   CALCULUS 

j^3     t^nnx-ni^nx^  whenx-  =  0.        Ans.     2. 

n  sin  X  —  sin  nx 

j^     ttinnx-ntmix^  when  n  =  1.        ^hs.     ^^ec^^^-tano:^ 

n  siu  a;  —  sin  nx  sm  a;  —  a;  cos  a; 


^ 


?)i  sin  :t-  —  sm  mx 


15. ,    when  x  =  0. 


(a;-l)e"  +  0«  +  l)e 

A" 


^ 


16.    ^^  +^ — ~^-^  ^   '^,  when  a- =  0.         Jbis.     8. 

■'^  log  sec^  a?  —  ar 

7^'  / 
98.     Evaluation  of  the  Indeterminate  Form  ^.      The  method  is  tlie 


,^   —     ^ 

same  as  that  given  in  Art.  97  for  tlie  form  - . 
It  has  been  shown  in  that  article  that 

Lim/,=o  ^  ■    ^  ^  (  =  ^-^ , (1) 

ij/{a  +h)      Y\Ci) 

if    </)(a)=0,    and    i/^(a)=0. 

It  may  be  shown  that  (1)  is  true  also,  if  (f>(a?)  =  go    and   i/r(a)  =  oo  . 
For  the  proof  of  this  the  student  is  referred  to  more  extensive 
treatises  on  the  Differential  Calculus. 

For  example,  find  the  limiting  value  of   ^^  ^ ,  when  x  =  0. 

cot  a; 

(b(x)        log.T         CO 

-t^^—i  =     '^     =  — ,  wdien  .T  =  0. 

lp(x)         cot  X         CO 

1 

di'(x)  X  sin^a;      0      -,  ^ 

^  ^  ^  = —  = =  -,  when  a;  =  0. 

i//'(a;)       —  cosec"a;  x  0 

^^^_2sin^cos^^0_.       whena;  =  0. 
xp"{x)  1  1 

Thus  the  required  limiting  value  is  0. 

CO  0 

N«iTE.  —  The  form  —  can  in  most  cases  be  avoided  by  transformation  into  — 

CO  -^  0 


IXDETKKMIXATK    FDKMS  HI 

Evaluation  of  the  Indeterminate  Forms  f )  •  x  ,  «  —  xi . 
Transform  the  expression  into  a  traction,  which  will  assume  either 

tlie  form  -  or  —  • 
0      :c 

For  example,  find  the  value  of 

(tt  —  2 X)  tan  x,  when  .c  =  ~. 
This  takes  the  form  0  •  cc  . 

But  (tt  -  2  a;)  tan  x  =  "^  ~  -  •^'  =  '^ ,  wh en  .»•  =  -  . 

-^--}-^  = —  J,  when  .(•  =     . 

if/  \x)       —  cosec-  X  2 

Thus  the  required  limiting  value  is  2. 

For  another  example,  find  the  limiting  value  of 

,  when  X  =  1. 

log  X      x  —  1 


This  takes  the  form  oo 


CO. 


T3      ,  1  1  X  —  1   —  log  X  0  1  ^ 

But = — 2— =  ^,  when  j-=  1 

log  a;     x—1      (x  —  1)  log  a;      0 

1-1 

iM  =  ^        =  2,  ,vhen  X  =  1. 

^'(-•)      1^1+log.     « 


^^  =  ^i_  =  l,when.r  =  l. 
^"(x-)       ]_^1      2' 

a^      X 
Thus  the  required  limiting  value  is    • 


112  DIFFERENTIAL   CALCULUS 

100.    Evaluation  of  the  Exponential  Indeterminate  Forms,  0",  1°°,  oo°. 

Take  the  logarithm  of  the  given  expression,  which  Avill  have  the 

form  -  or  —  •    The  limiting  value  of  this  logarithm  will  determine  the 

0      00 
given  function. 

For  exam])le,  find  the  limiting  value  of  a*'',    when  x  =  0. 
This  takes  the  form  0". 

Let  y  =  af; 

then  log  y  =  x  log  x  =     "  ^  —  — ,  when  x  —  0. 

^.-1  CO 

1 

r_L2  —— —  =  —  a;  =  0,  when  x  =  0. 
x^ 
Thus  the  limiting  value  of  log  y  is  0. 
Hence  the  limiting  value  of  y  is  e"  =  1. 

For  another  example  find  the  limiting  value  of 

(1  4-  ax)x,  when  x  =  0. 
This  takes  the  form  of  1*. 

Let  2/  =  (1  -|-  ax)x , 

,  log  (1  +  ax)      0      1  rt 

log  V  =     ^\    ^  ^  =  _   when  x  —  0. 


±M  =  l±^=:a,  whena-  =  0. 
The  limiting  value  of  log  y  being  a,  the  limiting  value  of  y  is  e". 


INDK!  i:i;mi.\ai  i;  loiiMS 


113 


EXAMPLES 

Find,    the    limiting    values  <if    the    f'ulldwiu!::  expressions   for  tlie 
assigned  values  of  the  variable. 


k:  1   1 


-log(l+x-), 


tan- 


X  tan  X  —  -  sec  .r, 


log  tan  ax 
log  tan  hx 


^- 


2  X-     2x  tan  a; 
sec  3  X 


sec  o  X 


^S.  (-^ 


2 

8.  (cosec^)'""" 

9.  (tan^)-'^^ 


11. 


O'^h-^if 


T  12.  <log  .t).-* 


14. 


OS  a.r  +  cos  hx 


when  a;  =  0. 
when  .r  =  00 , 
when  .1"  =  ^. 
when  .f  =  0. 
when  a;  =  0. 
when  .T  =  -• 

when  x  =  \. 
when  ^  =  ^- 
when  ^  =  -  • 
when  0  =  -. 

when  x-=0. 

when  X  =  e. 
when  .i;  =  0. 

when  X  =  0. 


^TIS. 

2' 

Ans. 

3 

Ann. 

-  1 

Aus. 

1 

Ajis. 

1 
0 

Ans. 

5 
3 

.-1;ks.  1. 

Ans.  e  3- 

^Ih.s.  abc. 

Ans.  €'. 

Aixs.  ae. 

Ans.  ^ 


CHAPTER   X 

MAXIMA    AND    MINIMA    OF    FUNCTIONS    OF    ONE 
INDEPENDENT   VARIABLE 

101.  Definition.  A  maximum  value  of  a  function  is  a  value  greater 
than  those  immediately  preceding  or  immediately  following. 

A  minimum  value  of  a  function  is  a  value  less  than  those  immedi- 
ately preceding  or  immediately  following. 

If  the  function  is  represented  by  the  curve  y=f(x),  then  PM 
represents  a  maximum  value  of  y  or  of  f{x),  and  QN  represents  a 
minimum  value. 


102.    Conditions  for  a  Maximum  or  a  Minimum. 

It  is  evident  that  at  both  P  and  Q  the  tangent  is  parallel  to  OX, 
and  therefore  we  have  for  both  maxima  and  minima, 


dy 
dx 


0. 


Moreover,  as  we  move  along 
the  curve  from  left  to  right, 
at  P  the  slope  changes  from 
positive  to  negative  ;  but  at  Q, 
from  negative  to  positive. 

In  other  words, 

At  P  the  slope  decreases  as 
X  increases.     ...         .     (a) 

At  Q  the  slope  increases  as 
X  increases (b) 


By  Art.  21  we  have  the  case  (a). 


when 


|(slope)<0. 
114 


al 


/  MAXIMA   AND   MINIMA    OF   FUNCTIONS  115 

But    ■  A(slope)  =  ^f^-^V^, 

and  (1)  becomes  ^  <  0. 

rfar' 

Hence  when  -'^  =  0,    and  ^^  <  0, m 

dx  djcr  ^  ' 

there  is  a  maximum  value  of  y. 

By  similar  reasoning  we  have  the  case  (6), 

when  —^  >  0. 

dxr 

Hence  when  ^^  =  0,   and  ^,  >  0, (3) 

dx  dxr 

there  is  a  minhmim  value  of  y. 

For  example,  let  us  find  the  maximum  and  miniinnm   values  of 


the  function 


^-2x^  +  ?>x  +  \. 


Put  w  =  ^-2.»r  +  3.7;-|-l. 


Then  ^  =  x--4a-  +  3,        (4) 

ax 

f?=2a:-4 (o) 

dx- 

Putting  (4)  =  0,  x2  -  4  .T  +  3  =  0, 

whence  x  =  l  or  3. 

Substituting  those  values  of  x  in  (;"5),  we  find 

whena;  =  l,  P:.= -^  <<^; 

ax- 
when  a- =3,     "  ''!^{=2>0. 


116  DIFFERENTIAL   CALCULUS 

Hence  by  (2)  and  (3), 


when  x==l, 
when  x  =  3, 


y  has  a  maxinmm  value : 
y  has  a  minimum  value. 


From  the  given  function  we  find 

that  the  maximum  value  oi  y  is     y  =  2^, 

and  the  minimum  value  oi  y,  ?/  =  1. 


103.    In  exceptional  cases  it  may  happen  that  a  value  of  x  given  by 


^  =  0,  makes  ^,: 
dx  dx- 

This  would  be  the  case  for  a 
point  of  inflection  li  (see  Art. 
158)  whose  tangent  is  parallel 
to  OX.  Here  the  ordinate  RL 
is  neither  a  maximum  nor  a 
minimum. 

But  there  may  be  a  maximum 
or   minimum   value  of  y,    even 

when  ^=0.     This  is  more  fully 
dx' 

considered  in  Art.  106.  The 
following  article  is  also  appli- 
cable to  such  cases. 


0,  so  that  neither  (2)  nor  (3),  Art.  102,  is  satisfied. 
Y 


104.    Second  Method  of  determining  Maxima  and  Minima.     Maxima 

and  minima  may  be  determined  from  the  first  derivative  —  alone, 

d-y  ^^ 

without  using   — ^ . 

We  have   seen   in  Art.  102  that  when  y  \s  a,  maximum,  as  at 


and  when  ?/  is  a  mini- 


P,  the  slope,  that  is,  — ,,  changes  from  +  to 

mum,  as  at  Q,  -^  changes  from  —  to  +  .     (It  is  understood  that  w( 
dx 

pass  along  the  curve  from  left  to  right.) 


MAXIMA    AND    .MINIMA    OF    irXCTIOXS  117 

By  examining  the  form  of  -•[,  which  shoukl  bu  expressed  in  factor 

form,  we  may  determine  whether  it  changes  from  •+•  to  —  .  or  f !•..!,) 
—  to  + ,  for  any  assigned  vahie  of  x. 
Let  us  apply  this  method  to  the  example  in  Art.  102, 

|  =  a;=-4a;+3  =  (x-l)(..--3). 

Here  —  can  change  sign  only  when  x  =  \  or  x  =  3 
dx  "^ 

By  supposing  x  to  change  from  a  value  slightly  less,  to  one  slightl 
greater  than  1,  we  find  that  (x  —  1)  changes  from  —  to  +  ;  but  sine 

the  factor  (a;  —  3)  is  then  negative,  it  follows  that  -^  changes  from 

dx 
+  to  — ,  when  x  =  1,  and  denotes  a  maxinuim.     In  the  same  way,  we 

find  that  -^  changes  from  —  to  +,  when  x  =  3,  and  denotes  a  miui- 


y 
since 


mum. 


Again,  consider  the  function  ?/  =  (x  —  4)^0;  +  2)^ 
Diiferentiating  and  writing  the  result  in  factor  form, 
g=3(3a--2X-c-4)Xx  +  2)l 

2  dii 

When  X  =  ",      -^  changes  from  —  to  -f- . 

3  dx 

When  x  =  —2,  —  changes  from  -f  to  —  • 
dx 

When  X  =  4,      ^-  •-  does  not  change  sign, 
dx 

since  (a;  —  4)*  cannot  be  negative. 

2 

Hence  we  conclude  that  y  is  a  mininuim  wlien  a;  =  ^;  a  maximum 

u 

when  x  =  —  2 ;  but  neither  a  maximum  nor  minimum  when  x  =  4. 

'      As   this   method   does  not  require  ^-^,,  it  is  preferable   to  that 

dx- 
of  Art.  102,  when  the  second  differentiation  of  //involves  much  work. 


118  DIFFERENTIAL   CALCULUS 

EXAMPLES 

1.  Find  the  maximum  value  of     32  a;  — a;*.  Ans.     48 

Find  the  maximum  and  minimum  values  of  the  following  functions 

2.  2a;''  —  3 a;^  —  12  x  +  12.       Aris.     x  =  —  1,  gives  a  maximum4p 

a;  =  2,  gives  a  minimum  4-[8 

2 

3.  2 a;^  —  11  aj^  +  12 a;  +  10.       Ans.     x  =  ^,  gives  a  maximum  13i-|^, 

o 

X  =  3,  gives  a  minimum       1 

3a-  •  9  a^ 

4.  af  -f  9  (a  —  a;)^  ^ns.      a;=  — -,  gives  a  maximum  — 

3 «      •  •    •  9  a^ 

a;  =  -^ ,  gives  a  minimum  

4  16 

5.  (a;-l)(a;-2)(x-3). 

1        .  .  2 

A71S.     X  =  2 ,  gives  a  maximum  — ~— 

VS  3V3 

1  •  ■   •  2 

a;  =  2  -| -,    gives    a    minimum 

V3     .  3V3 

6.  2  (3  a;  +  2)2  —  3  x*.  Ans.     x  =  2,  gives  a  maximum  80 

3.4 4x^4-8  a; 8 

7.  Show  that — — '- has  no  maximum  nor  minimum 

X—  1 

8.  ^  +  _^ — ,  where  a  >  6. 

^       ^^-^  .  a'         .  .  (a  +  bY- 

Ans.     X  = ,  gives  a  maximum-!^ — ■ — ^ 

a  +  b  a       ' 

a'         .  .    .  (a -by 

X  =■ ,  gives  a  minimum  ^^ '— 

a  —  b  a      ' 

9.  Show  that  the  greatest  value  of    — ^^-   is  — . 

x"  ne 

9 

10.  Show  that  the  greatest  value  of    cos  2  ^  +  sin  6    is  -. 


i 


MAXIMA   AND   MINIMA   OF   FUNCTIONS  110 

11.    Show  that  the  maximum  and  miiiimiim  values  of 


¥ 


iH 


sin^' e  +  sin- {[^-6]    are    ,^    and    ^. 


12.  rind  the  maximum  vahie  of    a  sin  x  +  b  con  x.  Ans.   ^/^^nf]^ 

13.  Find  the  maximum  vahie  of  tan"'  ar  —  tan"'  -,  the  angles  being 

Ins.     tan" 


taken  in  the  first  quadrant.  ,  'A 

Ano  frir>~l 


4- 

14.    Show  that  the  least  value  of   a- tan*  ^  +  6- cot- 6  is  the  same  as 
that  of   a*e"'  +  6^-"^,   and  equal  to  2  ab. 

^-3^15.   y  —  - -^.  Ajis.  a  minimum  when  x  = - 

a~2x  ,  4" 

f\6.   y  =  {x~l)\x  +  2f. 

Ajis,    a  maximum  when  .X' =  —  '- ;   a  minimum  when  x  =  l; 
neither  when  a;  =  —  2. 


17.   y=(x~2yX2x  +  l)\ 

Ans.    A  maximum  Avhen  x  = ;  a  minimum  when  x  =  — , 

2  1«' 

neither  when  a;  =  2. 


105.   Case  where  —  =  oo  .    It  is  to  be  noticed  that  —  may  change 
dx  dx 

sign  by  passing  through  infinity  instead  of  zero. 

Hence  if  ^  =co  , 

dx 

for  a  finite  value  of  it-,  this  value  should  be  examined,  as  well  as 
those  given  by 

dx 

I 


120  DIFFERENTIAL   CALCULUS 

For  example,  suppose 


Then 


y  —  a  —  h{x  —  cy 
V  2b 


hence  we  have 

-M  —cc       when  x  =  c. 
dx 


3(.i;-c)3 
Y 


It  is  evident  that  when  x  =  c, 


cly 
dx 


changes  from  -f  to  — ,  indicating  a 
maximum  value  of  y,  which  is  a. 

The    figure   shows   the   maximum 
ordinate    PM,    corresponding    to    a  -^ 

cusp  at  P. 

On  the  other  hand,  suppose     y  —  a—  h(x  —  c) s. 


\ 


Then 


dy 
dx 


3(x-c)' 


=  00    when  x  =  ( 


But  as  -^  does  not  change  sign  when  x  =  c,   there   is   no   maxi-       i 
dx  "^  f 

mum  nor  minimum.    The  corresponding  curve  is  shown  m  the  figure. 


MAXIMA    AM)   MINLMA   OF   FUNCTION'S  121 

EXAMPLES 

Find   the   maximum  and  minimum  values  of  tne  two  following 
functions: 

1.    y=(x  +  l)i{x-ny. 

Ans.  A  minimum  when  x  =  5;  a  maximum  when  x  =  ^: 
a  minimum  when  a;  =  — 1. 


> 


2.    ij  =  (2x~a)^x-ay' 


Alls.  A  maximum  when  x  =  ^;  a  minimu^m  when  x  =  a. 


106.  Conditions  for  Maxima  and  Minima  by  Taylor's  Theorem. 
Suppose  the  function /(.r)  to  be  a  maximum  when  x  =  a.  Then,  by 
the  definition  in  Art.  101, 

f{a)>f(a  +  h), 

and  also      "  f{ci)>f{a  —  h), 

where  h  is  any  small  but  finite  quantity,     Now,  by  the  substitution 
of  a  for  X  in  Taylor's  Theorem,  we  have 

f(a  +  h)-f(a)=     A/'(o)  +  |/"(a)  +  |/"'(a)  +  ....     .    (1)* 

/(a_7,)_/(a)  =  -7i/'(a)  +  |/''(o)-|/'''(«)  +  ---.     •     (-') 

By  the  hypothesis  /(a  +  h)  —f  («)  <  0, 

and  also  f{a  —  h)-f(a)<0. 

Hence  the  second  members  of  both  (1)  and  (2)  must  l)e  negative. 

*  The  rigorous  form  of  Art.  9.S  may  be  used  here  without  any  change  in  the 
context. 


122  DIFFERENTIAL  CALCULUS 

By  taking  h  sufficiently  small,  the  first  term  can  be  made  numeri- 
cally greater  than  the  sum  of  all  the  others,  involving  /i^,  h^,  etc. 
Thus  the  sign  of  the  entire  second  member  will  be  that  of  the  first 
term.  As  these  have  different  signs  in  (1)  and  (2),  the  second  mem- 
bers cannot  both  be  negative  unless 

/(a)  =  0. 

Equations  (1)  and  (2)  then  become 

7)2  7)3 

/(a  +  /0-/(a)  =  |/"(a)  +  |/'»+-, 

/(a-/0-/(a)  =  ^\r'(a)-^V"'(a)+-. 

The  term   containing  /r  now  determines  the  sign  of  the  second 
members.     That  these  may  be  negative,  we  must  have 
/»<0. 

If  then  /'(cO  =  0     and    /"(a)<0,  . 

/(a)  is  a  maximum. 

Similarly,  it  may  be  shown  that  if 

/(a)  =  0     and    /"(«)>  0, 
/(«)  will  be  a  minimum. 

If  /'(«)  =  ^     and    /"(a)=0, 

similar  reasoning  will  show  that  for  a  maximum  we  luust  also  have 

/"'(a)=0     and     /'»<0; 

and  for  a  minimmn 

/'"(«)=  0     and    f\a)>0. 

The  conditions  may  be  generalized  as  follows : 
Suppose  that 

/'(a)  =  0,    /"(a)  =  0,    /'"(a)  =  0,     •.•    f\a)  =  0, 

and  that/"^X'^0  ^'^  not  zero. 

Then  if  n  is  even, /(a)  is  neither  a  maximum  nor  a  minimum. 
If  n  is  odd, /(«)  will  be  a  maximum  or  a  minimum,  according  as 
/"+Xa)<0     or     >0. 


MAXIMA   AND   MINIMA   OF   FUNCTIONS 


128 


PROBLEMS   IN    MAXIMA  AND   MINIMA 

1.  Divide  10  into  two  such  parts  that  the  pnithict  of  the  square 
of  one  and  the  cube  of  the  other  may  be  the  greatest  possible. 

Let  X  and  10  —  a;  be  the  parts.  Then  x^  (10  —  xy  is  to  be  a  maxi- 
mum.    Letting  u  =  .^•^(10  —  x)'',  we  find 


du 
dx 


x-(4-x)(10-a-)2  =  0, 


from  which  we  find  that  ti  is  a  maximum  when 
required  parts  are  4  and  (3. 


Hence  the 


2.  A  square  piece  of  pasteboard  wliose  side  is  a  has  a  small 
square  cut  out  at  each  corner.  Find  the  side  of  this  square  that  the 
remainder  may  form  a  box  of  maximum  contents. 

Let  X  =  the  side  of  the  small  square.  Then  the  contents  of  the 
box  will  be  (a  —  2  x^x.     Representing  this  by  u,  we  find  that  n  is  a 


maximum  w 


hen 


which  is  the  required  answer. 


•  3.   Find  the  greatest  right  cylinder  that  can  1 
given  right  cone. 

'LetAD  =  a,DC=h. 

Let  X  =  DQ,  the  radius  of  the  base  of  the  cylinder,  and  y  =  PQ, 
its  altitude. 

From  the  similar  triangles  ADC,  PQC, 
we  find 


y  =  l{b-x). 


The  volume  of  the  cylinder  is 
TTX^y  =  IT-  XT  {b  —  x). 

0 

This  will  be  a  maximum  when  u=h3c^—x^ 
is  a  maximum. 

This  is  found  to  be  when  x  =  lb,  the  radius  of  the  base  of  th 
5   required  cylinder. 

From  this,    y  =  ^,    the  altitude  of  the  cylinder. 
o 


124 


DIFFERENTIAL   CALCULUS 


4.    Detei-mine  the  right  cylinder  of  the  greatest  convex  surface 
that  can  be  inscribed  in  a  given  sphere. 

Let  /•=  OP,  the  radius  of  the  sphere;  x=  OR,  the  radius  of  base 
of  cylinder ;  and  y  =  PR,  one  half  its  altitude. 

From  the  right  triangle  OPR  we  have 

.^•2  +  2/2=  r". 
The  convex  surface  of  the  cylinder  is 

2  TTic  •  2 II  =  4  ttx  Vr-  —  x\ 

We  may  put  n,  equal  to  this  expression, 
and  determine  the  value  of  x  that  gives  a 
maximum  value  of  u.  But  the  w^ork  may 
be  shortened  by  the  following  considerations  : 


4  TTX  V/"  —  x^    is  fi 


J  a  maximum, 


when 

and 

when  its  square 


X s/r'-^  —  or    is  a  maximum ; 


*■    Hence   we   may   put  k, 

maximum,  when  x  = 

V2 

_    ^' 

^~V2' 

2y  =  rV2. 


xVr^  —  x-    is  a  maximum, 
r-x^  —  X*    is  a  maximum.* 

r-x"  —  X*,   from  which   we   find   w   is   a 


From  this 


givm 


g  for  the  altitude  of   the  cylinder, 


Another  Method.     The  equations 

The  convex  surface  =  4=.Trxy,    u  =  xy,    .     .     .     .     .     .     .     .    (1) 

^  +  f  =  A (2) 

may  be  used  without  substituting  in  (1)  the  value  of  y  from  (2). 

*  Since  we  are  only  concerned  with  the  positive  root  of  Vr^  —  ofi. 


MAXIMA    AND    MINIMA    OF    FUNCTIONS  126 

Differentiating  (1),  ^^,f^j.']l (^\ 

dx     •  ■>■■  ^  ^ 

and  differentiating  (2),    a;  +  ?/  ^  =  0, 

Substituting  in  (3), 

Since  x  and  y  are  positive  quantities,  it  is  evident   that  when 
x=y,  —  changes  from  +  to  — ,  giving  a  maximum  value  of  m. 


clu 

=  .'/- 

1     r'^!' 

dx 

'      dx 

dy_ 
dx 

=  0, 

d!f 
dx 

X 

y 

du_ 
dx 

■-I/- 

y 

y-- 
y 

ar 

Cojribining  ^  =  y,   with  (2),  we  have 


X  =  — -,  y  =  — ^ ,  as  before. 

V2  V2 

In  some  problems  this  method  has  some  advantages  over  the  first. 

A- 

^  5.  Divide  48  into  two  parts,  such  that  the  sum  of  the  square  of 
one  and  the  cube  of  the  other  may  be  a  minimum.         Ans.   42|,  5^. 

6.  Divide  20  into  two  parts,  such  that  the  sum  of  four  times  the 
reciprocal  of  one  and  nine  times  the  reciprocal  of  the  otlier  nni}-  l)o 
a  minimum.  Ans.   8,  12. 

'^'  7.  A  rectangular  sheet  of  tin  15  inches  long  and  8  inches  wide 
has  a  square  cut  out  at  each  corner.  Find  the  side  of  this  square 
so  that  the  remainder  ma/  form  a  box  of  maximum  contents. 

Ans.   \\  in. 

8.  How  far  from  the  wall  of  a  house  must  a  man,  whose  eye  is 
5  feet  from  the  ground,  stand,  so  that  a  window  5  feet  high,  whose 
sill  is  9  feet  from  the  ground,  may  subtend  the  greatest  angle  ? 

Ans.    C  ft. 

9.  A  wall  27  feet  high  is  8  feet  from  the  side  of  a  house.  What 
is  the  length  of  the  shortest  ladder  from  the  ground  over  the  wall 
to  the  house?  Ana.    13\^i;i=  10.87  fi. 


126  DIFFERENTIAL   CALCULUS 


H 


10.  A  person  being  in  a  boat  5  miles  from  the  nearest  point  of  the 
beach,  wishes  to  reach  in  the  shortest  time  a  place  5  miles  from  that 
point  along  the  shore ;  supposing  he  can  run  6  miles  an  hour,  but 
row  only  at  »the  rate  of  4  miles  an  hour,  required  the  place  he  must 
land.  Ans.   929.1  yards  from  the  place  to  be  reached. 

11.  Find  the  maximum  rectangle  that  can  be  inscribed  in  an 
ellipse  whose  semiaxes  are  a  and  h. 

Ans.    The  sides  are  «V2  and  h\/2;  the  area,  2  ah. 

12.  A  rectangular  box,  open  at  the  top,  with  a  square  base,  is  to 
be  constructed  to  contain  500  cubic  inches.  What  must  be  its 
dimensions  to  require  the  least  material  ?  • 

Ans.    Altitude,  5  in  ;  side  of  base,  10  in. 

"•^  13.  A  cylindrical  tin  tomato  can  is  to  be  made  which  shall  have 
a  given  capacity.  Find  what  should  be  the  ratio  of  the  height  to 
the  diameter  of  the  base  that  the  smallest  amount  of  tin  shall  be 
required.  A)ts.    Height  =  diameter. 

V  14.  What  are  the  most  economical  proportions  for  an  open  cylin- 
drical water  tank,  if  the  cost  of  the  sides  per  square  foot  is  two 
thirds  the  cost  of  the  bottom  per  square  foot  ? 

Ans.    Height  =  |  diameter. 

15.  (a)  Find  the  altitude  of  the  rectangle  of  greatest  area  that 
can  be  inscribed  in  a  circle  whose  radius  is  r. 

Ans.   ?•  V2 ;  a  square. 

(p)  Find  the  altitude  of  the  right  cylinder  of  greatest  volume  that 

2  r 
can  be  inscribed  in  a  sphere  whose  radius  is  r.  Ans.   -^^^— • 

V3 

16.  (a)  Find  the  altitude  of  the  isosceles  triangle  of  greatest  area 
inscribed  in  a  circle  of  radius  r.  Ans.    ^ ;  equilateral  triangle. 

(h)  Find  the  altitude  of  the  right  cone  of  greatest  volume  inscribed 
in  a  sphere  of  radius  r.  Ans.    —  • 


MAXIMA    AND    .MINIMA    OF    FUNCTIONS  liiT 

17.  (a)  Find  the  altitude  of  the  isosceles  triangle  of  least  area 
circumscribed  about  a  circle  of  radius  i\ 

Ans.   3r;  equilateral  triangle. 
(b)  Find  the  altitude  of  the  right  cone  of  least  volume  circum- 
scribed about  a  sphere  of  radius  /•.  ^l)(,s>.   4  ,■ 

18.  A  right  cone  of  maximum  volume  is  inscribed  in  a  given  right 
cone,  the  vertex  of  one  being  at  the  center  of  the  base  of  the  other. 
Show  that  the  altitude  of  the  inscribed  cone  is  one  third  the  altitude 
of  the  other. 

A 

'^    19.    Find  the  point  of  the  line,   2a;  +  y  =  16,   such  that  the  sum 

■  of  the  squares  of  its  distances  from  (4,  5)  and  (6,  —  3)  may  be  a 

minimum.  Ans.    (7,  2). 

&^ 

V"       20.    Find  the  perpendicular  distance  from  the  origin  to  the  Hue 

+  ■'  =  1,   by  finding  the  minimum  distance.  Ans. 


ah  ^d'  +  U' 

21.  A  vessel  is   sailing  due  north   10  miles  per  hour.     Another 
vessel  190  miles  north  of  the  first  is  sailing  15  miles  per  hour  on  a 
course  East  30°  South.     When  will  they  be  nearest  together,  and* 
what  is  their  least  distance  apart  ?  

Ans.    In  7  hrs.     Distance  15Vo7  =  113.25  mi. 

22.  A  vessel  is  anchored  3  miles  off  the  shore.  Opposite  a  point 
5  miles  farther  along  the  shore,  another  vessel  is  anchored  9  miles 
from  the  shore.  A  boat  from  the  first  vessel  is  to  land  a  passenger 
on  the  shore  and  then  proceed  to  the  other  vessel.  What  is  the 
shortest  course  of  the  boat?  Ans.    13  mi. 

23.  The  velocity  of  waves  of  length  A.  Ih  deep  water  is  propor- 
tional to  \/-  +  - ,    where  a  is  a  certain  linear  magnitude.    Show  tliat 

><  a     X 
the  velocity  is  a  minimum  when  X  =  a. 

E 

24.  Assuming  that  the  current  in  a  voltaic  cell  is  C=  -— -jj*   -f- 

being  the  electromotive  force,  r  the  internal,  and  R  the  extvrnal. 
resistance;  and  that  the  power  given  out  \s  P=  R(J^\  show  that  /' 
is  a  maximum  when  It  =  r. 


128  DIFFERENTIAL   CALCULUS 

25.  Froia  a  given  circular  sheet  of  metal,  to  cut  out  a  sector,  so 
that  the  remainder  may  form  a  conical  vessel  of  maximum  capacity. 

Ans.   Angle  of  sector  =  ( 1  —  ^- )2  tt  =  66°  14'. 

26.  Find  the  height  of  a  light  on  a  wall  so  as  best  to  illuminate 
a  point  on  the  floor  a  feet  from  the  wall ;  assuming  that  the  illumi- 
nation is  inversely  as  the  square  of  the  distance  from  the  light,  and 
directly  as  the  sine  of  the  inclination  of  the  rays  to  the  floor. 

Ans. 

V2 

27.  At  what  point  on  the  line  joining  the  centres  of  two  spheres 
must  a  light  be  placed,  to  illuminate  the  largest  amount  of  spherical 
surface? 

Ans.  The  centres  being  A,  B;  the  radii,  a,  b;  and  Pthe  required 
point ;  AJ^~ :  FB'=  a' :  h^ 

*    28.    (a)  The  strength  of  a  rectangular  beam  varies  as  the  breadth 
and  the  square  of  the  depth.     Find  the  dimensions  of  the  strongest 
beam  that  can  be  cut  from  a  cylindrical  log  whose  diameter  is  2  a. 
Ans.   Breadth  =  —  .     Depth  =  2a^/2. 

(6)  The  stiffness  of  a  rectangular  beam  varies  as  the  breadth  and 
the  cube  of  the  depth.  Find  the  dimensions  of  the  stiffest  beam 
that  can  be  cut  from  the  log.  Ans.   Breadth  =  a.     Depth  =  a  \/3. 

29.  The  work  of  propelling  a  steamer  through  the  water  varies 
as  the  cube  of  her  speed.  Find  the  most  economical  speed  against 
a  current  running  4  miles  per  hour.  Ans.    6  mi.  per  hr. 

30.  The  cost  of  fuel,  consumed  in  propelling  a  steamer  through 
the  water  varies  as  the  cube  of  her  speed,  and  is  $25  per  hour  when 
the  speed  is  10  miles  per  hour.  The  other  expenses  are  f  100  per 
hour.     Find  the  most  economical  speed. 

Ans.  -^2000  =  12.6  mi.  per  hr. 

31.  A  weight  of  1000  lbs.  hanging  2  feet  from  one  end  of  a  lever 
is  to  be  raised  by  an  upward  force  at  the  other  end.  Supposing  the 
lever  to  weigh  10  lbs.  per  foot,  find  its  length  that  the  force  may  be 
a  minimum.  Ans.  20  ft. 


MAXIMA   AND   MINIMA   OF   FUNCTIONS  12!) 

•^' 

32.  (a)  The  lower  corner  of  a  leaf,  whose  width  is  a,  is  folded 
ovei'  so  as  just  to  reach  the  inner  edge  of  the  page.  Fincflhe  widtli 
of  the  part  folded  over,  when  the  length  of  the  crease  is  a  minimum. 

Ans.     \a. 
(b)  In  the  preceding  example,  find  when  the  area  of  the  triangle 
folded  over  is  a  minimum.  Ans.     AVhen  the  width  folded  is  I  u. 

.i 

'33.  A  steel  girder  25  feet  long  is  moved  on  rollers  along  a  pas- 
sageway 12.8  feet  wide,  and  into  a  corridor  at  right  angles  to  the 
passageway.  Neglecting  the  horizontal  width  of  the  girder,  how- 
wide  must  the  corridor  be,  in  order  that  the  girder  may  go  around 
the  corner?  Ans.     5.4  ft. 

34.  Find  the  altitude  of  the  least  isosceles  triangle  that  can  be 
circumscribed  about  an  ellipse  whose  semiaxes  are  a  and  b,  the  base 
of  the  triangle  being  parallel  to  the  major  axis.  Ans.  3  b. 

35.  A  tangent  is  drawn  to  the  ellipse  whose  semiaxes  are  a  and 
b,  such  that  the  part  intercepted  by  the  axes  is  a  minimum.  Show 
that  its  length  is  a  +  b. 


CHAPTER   XI 


PARTIAL   DIFFERENTIATION 


107.  Functions  of  Two  or  More  Independent  Variables.  In  the  pre- 
ceding chapters  differentiation  has  been  applied  only  to  f nnctious  of 
one  independent  variable.  We  shall  now  consider  functions  of  more 
than  one  variable. 

Let  u=f(x,ij) 

be  a  function  of  the  two  independent  variables  x  and  y. 

Since  x  and  y  are  independent  of  each  other,  we  may  suppose  x  to 
vary  while  y  remains  constant,  or  y  to  vary  while  x  remains  con- 
stant ;  or  we  may  suppose  x  and  y  to  vary  simultaneously.  We 
must  distinguish  between  the  changes  in  u  resulting  from  these  dif- 
ferent suppositions. 

Let  A^'M  denote  the  increment  in  ti  resulting  from  a  change  in  x 
only,  and  A^,?*  the  increment  in  ii  from  a  change  in  y  only. 

Let  Am,  called  the  total 
increment   of  n,    be   the    in-       ^  N  p' 

crement  when  x  and  y  both 
change. 

Suppose  u  the  area  of  a 
rectangle  whose  sides  are  x 
and  y. 

Then       u  =  xy. 

If  X  changes  from  OA  to 
OA',  while  y  remains  con- 
stant, u  is  increased  by  the  rectangle  AM. 


B 
B 

Ay 

AyU 

c 

y 

u 

X 

A.w 
Aa; 

0 

t 

\          A 

That  is, 


A^u  =  area  AM. 
130 


PARTIAL   DIFFERENTIATIOX  181 

If  y  changes  from  OB  to  OB',  wliile  x  remains  constant,  u  is 
increased  by  the  rectangle  BN. 

That  is,  A^//  =  area  BN. 

If  X  and  y  both  change  together,  we  have  for  the  t;ptal  increment 
of  u,  A«  =  area  xiM-\-  area  BN-\-  area  MN. 

108.    Partial  Diiferentiation.   This  supposes  only  one  of  the  inde- 
pendent variables  to  vary  at  the  same  time,  so  that  the  differentia- 
tion   is    performed    by  the    same  rules  that  have  been  applied  to  " 
functions  of  a  single  variable. 

If  we  differentiate  u=f{x,  y),  supposing  x  to  vary,  y  remaining 

constant,  we  obtain  —  • 
dx 
If  we  differentiate,  supposing  y  to  vary,  x  remaining  constant,  \ve 

obtain  —  • 
ay 

The    derivatives,    — ,    — ,  thus  derived,  are  called  partial  deriiXi- 
dx     dy 

ia'es,  and  a  special  notation,   --,    -— ,    is  used  for  them. 
dx      dy 

For  examj)le,  if  u  =  x^  +  2  x-y  —  f, 

—  =  3  X-  -f-  4  xy,     the  a-derivative  of  n. 
dx 

_  =  2  or  —  3  ?/^,      the'  ^/-derivative  of  u. 
dy 

In  general,  whatever  the  number  of  independent  variables,  the 
partial  derivatives  are  obtained  by  supposing  only  one  to  vary  at  a 
time. 

EXAMPLES 

Derive  by  partial  differentiation  the  following  results: 

rru  dii   ,      du 

•      1       oi  — -EH—,  X—^  +  V —  =  M. 

x  +  y  ox         dy 

2.   z  ={ax'  +  2bxy  +  qf)%  (px  +  cy) -^^  =  (ax  +  by) ~^. 


132  differp:ntial  calculus 

3.    u  =  (y-z)(z-a:)(x-y),  5^  +  5^+5^  =  ^- 


-  , dr         dr 


5.    u  =  log(x'  +  ax-y+bxy'  +  cy'),         x-^  +  y~  =  Z. 


6. 

rt 

z      X      y 

* 

7. 

w 

e^  +  e" 

% 

8. 

tt 

2tan- 

1^, 
2/ 

ax         ay         az 


dx      dy 


5?/-      ,  du      ,  4:X^ 


9.   z=(x  +y)(x'  -  fy,  y^+x^^  =  z. 


.   dx         By      X-  +  ?/^ 

^2  -I-  -•  — 
6a;         (9^ 


10. 


-        1  /^^Y4-  /^^'V4-  ^^'Y=       1 

''     (.^.  +  ^2  _^  ,2)i'       VW     ^52/^  "^  ^a.y     (x-'  +  f  +z^f 


11.   2  =logyX  +  log^2/;  a;  log  a; ^4-2/ log  2/^  =  0. 


12.    u=  e-'  sin  ?/  +  e"-  sin  x, 


il'y+  /'^'Y=  e2x  +  e2^  +  2  e^+^sin  (a;  +  y). 


13.   ?(  =  log  (tan  X  +  tan  y  +  tan  2), 


>^  +  sin2./^  +  sin2.^=2. 
dx  '  dy  dz 


PAIITIAL    DIFFKKKXTIA'rroX 


183 


109.    Geometrical  Illustration  of  Partial  Derivatives.    TiCt  z  =  j\x,  y) 
be  the  equation  of  the  surface 
APCIL 

The  ordinate  PN  is  thus 
given  for  every  point  N  in  the 
plane  XY. 

Let  APB  and  CPU  be  sec- 
tions of  the  surface  by  i)lanes 
through  P,  jiarallel  to  XZ  and 
YZ,  respectively. 

If  X  and  y  both  vary,  P 
moves  to  some  other  position 
on  the  surface. 

If  X  vary,  y  remaining  con- 
stant, P  moves  ou  the  curve  of  intersection  APB. 


Hence  —  is  the  slope  of  the  curve  APB  at  P. 
dx 

If  y  var}-,  x  remaining  constant,  P  moves  on  the  curve  CPD. 
Hence  —  is  the  slope  of  the  curve  CPD  at  P. 


110.  Equation  of  Tangent  Plane.  Angles  with  Coordinate  Planes. 
In  the  figure  of  the  preceding  article,  let  /-•  he  the  i)oint  {x',  y',  z') ; 
PT,  the  tangent  to  APB  in  the  plane  APXM;  and  PT,  the  tangent 
to  CPD  in  the  plane  CPNL. 

It  is  evident  from  the  preceding  article  that  the  otpiations  of  PT 
are 


dz'  ' 

z  —  z'  =  —  (x — x'),   y  =  y', 
dx 


and  of  PT', 


du 


,(u-y'),    ^  = 


(2) 


dx' '     dy' 


denote  the  values  of 


dz     clz 
dx'  dy 


,    reopectively,  for  (x'.  y',  :'). 


134  DIFFERENTIAL   CALCULUS 

The  plane  tangent  to  the  snrfa£e  at  P  contains  the  tangent  lines 
PT  and  PT'.     Its  equation  is 

.-.■  =  |;(x-.')  +  |(,-y);.   .   .  ,   .   (3) 

For  (3)  is  of  the  first  degree  with  respect  to  the  current  variables 
a;,  y,  z,  and  is  satisfied  by  (1),  and  also  by  (2). 

TJie  equations  of  the  normal  through  P  are  those  of  a  line  through 
(x',  y',  z')  perpendicular  to  (3).     Its  equations  are 

x  —  x'     ?/  —  •?/'           .         ^  ... 

-  -  -(2-^0 (4) 


dz'  dz^ 

'  dx'  dy' 

TJie  angles  made  by  the  tangent  x)lane  ivitJi  the  coordinate  iiilanes  are 
equal  to  the  inclinations  of  the  normal  to  the  coordinate  axes. 

By  analytic  geometry  of  three  dimensions,  the  direction  cosines  of 
the  line  perpendicular  to  (3)  are  proportional  to 

Bz'      dz'        -. 
dx''    dy'' 

Hence,  if  a,  jS,  y,  are  the  inclinations  of  the  normal  to  OX,  OY, 
OZ,  respectively, 

cos  a  _  cos  /3  _  cos  y 
d£  ~  dz[  ""^TX 
dx  By' 


(5) 


Also  cos^  a  +  cos^  ;8  +  cos- y  =  1 (6) 


From  (5)  and  (6)  we  find,  dropping  the  accents, 

.dx) 
CDS'' a  = 

1 


-(ijKir 


PARTIAL   DIFFEllEXTIATJON  135 


cos-  y : 


^HUHI 


^HlJ-d 


For  the  inclination  of  the  tangent  plane  to  XY,  we  have  from  (7), 
and  Un^y=:fi'y+(^' 


The  term  slope  used  in  geometry  of  two  dimensions  may  thus  be 
extended  to  three  dimensions,  as  the  tangent  of  the  angle  made  by 
the  tangent  plane  with  the  plane  XY.     In  this  sense, 


the  s 


--<^^-(^T 


EXAMPLES 

1.    Find  the  equations"  of  the  tangent  phine  and  normal,  to  the 
sphere     ar  +  y-  -^z-  =  cr,    at  (x',  y,'  z'). 

dx  ;       z\'     dy  z' 

T^  dz'  x'     dz'  y' 

Hence  — ;  = :,    -r—  =  —-^' 

dx  z      Oy'  z 

^     Substituting  in  (3),  ^  -  z'  =  -  ^  {x-x')  -  -^  (y  -  y'), 

aa'  +  .y.v'  +  ::-z'  =  x'2  -f  »/'-  +  2 *=  =  a-.    An s. 


136  DIFFERENTIAL   CALCULUS 

From  (4)  we  find  for  the  normal 

x'  y 

^  _  1  —  ?'-  _  1  —  ?.  _  1  -^  —  y  —  ?.       .1 

x'  y'  z'  x'      y'      z' 

2.  Find,  the  equations  of  tangent  plane  and  normal  to  the  cone, 

3  X'  —  y-  +  2z-  =  0,    at  (x',  y',  z'). 

Ans.    3  XX'  -  yif  +  2  zz^  =  0,        ^^11^'  =  -Ll^  =  ^^  • 

3  x'  —y'         '2z' 

3.  Find  the  equation  of  the  tangent  plane  to  the  elliptic  parabo- 
loid,     z=  3  X-  +  2 1/,    at  the  point  (1,  2,  11). 

Ans.    6x-+ 8?/  — 2  =  11. 

4;.    Find  the  equations  of  tangent  plane  and  normal  to  the  ellipsoid, 
x'  +  2y''  +  3z'  =  20, 
at  the  point    x  =  3,    y  =  2,    z  being  positive. 

Ans.    3 A- +  4?/ +  32  =  20;    x==z  +  2,    3y^4z  +  2. 
Find  the  slo2)e  of  this  tangent  plane.  •  '  Ans.   |. 

5.    Find  the  equation  of  the  tangent  plane  to  the  sphere, 

x^  +  y-  +  z-  —  2x  +  2y  =  1,    at  (.(-■',  y',  z'). 

A71S.    xx' +  yy' +  zz'--x-x' -{-y  +  y'  =  l. 
^.   '.    ,       • 
111.    Partial  Derivatives  of  Higher  Orders,     By  successive  differ- 
entiation, the  independent  variables  varjnng  only  one  at  a  time,  we 

may  obtain 

5^    a^    a^    6%  _ 

dx^'  a/  dx''  a/"** 

If  we  differentiate  ?/.  with  respect  to  x,  then  this  result  with  respect 

to  y,  we  obtain  —  ( — '■],  which  is  written  ■ . 

dy\dxj  dydx 


PARTIAL    DIFFKKKN  riA'I'IOX  137 

Similarly,  ^    ,  .,  is  the  result  of  three  suecessivt!  (lifffrentiatioiis, 
dydx- 

two  with  respect  to  x,  and  one  with  respect  to  y.    It  will  now  l)e  shown 

that  this  result  is  independent  of  the  order  of  these  differentiations. 

In  other  words,  the  operations  —  and  —  are  coninuitativc . 
dx  dij 

,„,    ,    •         5-//  d-i(  d^ii  d^ii  ff^it 

That  IS, =  — -jr-,     =  ■ = . 

dtjdx       dxpy      dydx'-      dxdydx       dx'-dy 

* 
112.    Given  u=f(x,y), (1) 

to  prove  that  T"   T"  H  ^    n 

Supposing  X  to  change  in  (1),  y  being  constant, 

^''Xx~'  I^  ^"^ 

} 

Now  supposing  y  to  change  in  (2),  x  being  constant, 

A  f^^  =  f^-''  +  ^■*''  y  +  ^-'^  -/(-r,  y  +  A//)  -/(.«  +  A.r,  ?/)  +f(x,  y)^ 
Ay\^xJ  A//A.f 

Keversing  the  above  order,  we  find 

A^  ^  f(x,  y  +  Ay)  -f(x,  y)     ^^^ 
Ay  Ay 

Af^i^^  f(^  +  ^^'^  -I  +  ^"^  ~f^^  +  ^^''  y^  ~  •^^^'  -^  "*"  ^^^  "^  ^^^'  •^^'. 
A.K\A?/y  A.r  Ay 


-f-;).A(^l <'-'^ 


Ay\AxJ      Ax\Ay, 


The  mean  value  theorem,  (2),  Art.  90,  may  be  expressed   in    tl 

form     ^'=  f'(x+e-Ax),     where  «=/(x).      0<5<1. 
Ax 


138  DIFFERENTIAL   CALCULUS 

In  the  preseii-t  case,  where  u  =f(x,  y), 

Ax 


l~)=^^fA^  +  Or^x,y)^fJx-]-drAx,y  +  e,.Ay). 

\AxJ      Ay 


Ay 


Similarly,      —  =  fJx,  y  +  O-^-Ay), 

and  ^/^\  =  fJx  +  e,-Ax,  y  +  ^,. Av). 

Ax\AyJ 

By  (3)  fy,{x+6,-Ax,  y  +  e,-Ay)=f,lx  +  6,- Ax,  y  +  6,-Ay). 

Taking  the    limits  as  Ax,  Ay,  approach  zero,  and   assuming  the 
functions  involved  to  be  continuous. 


fyX^,y)=f.y{->',y)- 

That  is,  —  —   =—  — ),  or = -, 

oy\dxJ     ax\dyj  dydx      dxdy 


This  principle,  that  the  order  of  differentiation  is  immaterial,  may 
be  extended  to  any  number  of  differentiations. 


Thus, 


dhi  ^   a-  (du\  ^   d^   fdu 


dydxr      dydx\dxj  dxdy\dxj       dxdydx 

^  d  r  d-u  \  ^  /  d-H  \        d-n 

•                                dx\dydxj  dx\dxdyj      dx'-dy 

It  is  evident  that  the  same  is  true  of  functions  of  three  or  more 
variables. 

*/x(x,  y)  =  -f /(x,  y),  M^^,  y)  =  f  /(a;,  y), 

dx  By 


TARTIAL    DIFFERENTIATION 


139 


EXAMPLES 


XT     •£       d'u  d'-K  ,,         1    ., 

Verity = m  Lxs.  1-3. 

ox  a;/      0>j  ax 


1.    n  = 


ax  +  hii 


2.    u  =  a-^  log" 


ay  +  hx 
Derive  the  following  results: 
4.    H  =  ax^  4-  G  bx'-if-  +  c?/*, 

\^      5.    (t  =  log  (x^+rf), 

e.    z=  (3.C  +  iiy+  sin  (2x  -  ?/), 

7.  ■«  =  -  +  log  -  , 

y         II 

8.  z  =  X-  tau~^  •  —  y-  tan  ^  - , 

X  y 

9.  q—  (?•"  +'/•' ")  cos  nd, 
-  10.    «  =  log  (e^+e"+e-')> 

11.  ?t  =  ztan~^-, 

2/ 

12.  M  =  log(ar^+?/2  +  z^, 

XV' 

13.  M  =  ?/-2;-e-  +  2-are-  +  ar^'e-, 


3.    ?<  =  (.»  +  //)(''- 


ast 


0^/ 


a^u 

dxrdy-       dxdydxdy       dxdy'-dx  ' 

d'-i(      d'-a ^ 

dxr      dy- 

find   — , 0  -^ . 

d.c-      dx  by         dy- 

find  .,^-+2^  _+,._ 

S-z  _  •'^  —  y' 

dxdy       x~  +  y-^ 

^  4. 1  ^  +  1  i'v  ^  0 


d^u 


0,ji+!/+z-3u 


dxdydz 


dj?      dy-      dz- 


a«/< 


djrdy'dr 
14.   w  =  sin  (y  +  ^)  sin  (z  +  x)  sin  (x  +  ?/), 


=  t^-  +  e-'  4-  e'. 


.i? 


dxdydi 


=  2co8(2ir  +  2y  +  22). 


140  DIFFERENTIAL   CALCULUS 

113.  Total  Derivative.  Total  Differential.  In  Art.  107  we  have 
referred,  to  the  change  in  w  when  x  and  ;j  vary  siuiultaneousiy. 
This  change  is  called  the  total  increment  of  u.  Thus  the  total  lucre- 
"^entof  u=f{x,y)   - 

is  All  =  f{x  +  Ax,  y  +  Aij)-  f(x,  ?/). 

The  terms  total  derivative  and  total  differential  are  also  used.     For 

example, 

let  n  =  x'y-3xY, (1) 

and  suppose  x  and  y  to  be  functiorLs  of  a  variable  t. 

Differentiating  with  refe^ct  to  t^^^^^Cj^  v 

du       d  (a  ,        d  ^.^  o  Us 

1        'i  dii  ,   n   o    dx      ^    o    rfv      (,     o  dx 
dt  dt  dt  dt 

•/      .t'  /r,     0  n        •>\   dx    ,     /    ^         o     •>    \dy  /o\ 

^y  =(3x-,-6.r)^  +  (^--6-i/)^ (2) 


^ 


But  from  (1)  we  find 


—  =3  x-y  —  6  xir,         —  =  x*  —  Gary. 
dx  -^  •^'         dy 

So  that  (2)  may  be  written 

du  _  du  dx      du  dy'  ■  '      ro\ 

dt~dxdt       'dy  dt'    ''''''     ^    ^ 

If  we  had  used  differentials  in  differentiating  (1)  we  should  have 
obtained 

du  =  -^'  dx  +  ^  dy (4) 

dx  dy 

—  in  (2)  and  (3)  is  called  the  total  derivative,  and  du  in  (4)  the  total 

differential,  of  u. 

We  proceed  to  show  that  (3)  and  (4)  are  true  for  any  function  of 
X  and  y. 


I'AR'l'lAl.    DIKFKKKMIAIION  141 

Noticing  that  A(t  is  the  total  increment  of  u, 
and  Aji,  AyU,  the  partial  increments,  when  x  ami  >/  vary  separateh  , 
let 

«  =J{-^y!/)>  X  and  y  being  functions  of  /. 

M'=/(.r  + A.r,.v), 

u"  =J\x  +  Ax,!/  +  ^y). 

Then  A^n  =  *<'  —  u, 

A,/<'  =  /^"  —  a', 
A  »  =  (<"  —  n. 

Hence  ^n  =  A^n  +  A,,w', 

,  Ak      a,?/  Ax'  ,   A,,»<'  Am 

and  -     =  -^^ — " -. 

At       Ax    At       A;/     At 

Taking  the  limits  of  eacli  memher,  as  At.  and  consequently  Ax,  A;/, 
approach  zero,  du  ^du  (Jx      d.  cly  .. 

'  dt       dx  dt       (>y  dt' 
since  the  limit  of  u'  is  ?<•  ^ 

This  may  be  written  in  the  differential  form 

du  =  ^ldx^^^dy (6) 

ox  ay 

In  the  same  way,  if  u  =f{x,  y,  z),  where  x,  y,  z,  arc  functions  of  ^ 

we  find  du^dudx      dn(l>i_^clu(h ^^. 

dt       dx  dt       dy  dt       dz  dt* 

and  au  =  ^^dx  +  ^^dy  +  p^dz (8) 

dx  ay  dz 

We  may  write  in  (8) 

£*-"-•"       I'"-"'"'       l;''^="="' 

giving  du  =  dj.n  +  d,/t  -f  djt, 

that  is,  the  total  differential  of  it  is  the  sum  of  its  jKirtial  differentinla 


142  diffp:rential  calculus 

This  principle,  as  expressed  by    da  =  d^u  +  d^u,    may  be  illustrated 
by  the  tigure  of  Art.  107,  from  which  we  have 

Ay*  =  A^?(  +  A,^w  +  area  MN, 

that  is,  A?t  =  A_j.v6  +  A^»  +  Ax  A_?/. 

As  Ax  and  A//  approach  zero,  the  last  term  diminishes  more  rap- 
idly than  the  others,  and  we  may  write 

A«  =  A^y<  +  A^?«,  approximately, 

the    closeness    of    the   approximation   increasing    as    Ax    and    A?/ 
approach  zero. 

If  in  (5)  we  suppose  t  =  x, 
then  u  =/(x,  y),  y  being  a  function  of  x ; 

and  (5)  becomes  da^du^dud^ ^^^ 

dx      'dx      ay  dx 

Similarly,  if  in  (7),  t  =  x, 

u  =/(x,  y,  z),        y  and  z  being  functions  of  x; 

,        „                            du      du  .   da  dy  ,   dti  dz  ,^  ^. 

whence  =       +       ^-|_  (10) 

dx      ox      dy  dx      dz  ax 


EXAMPLES 

Find  the  total  derivative  of  u  by  (5)  or  (7)  in  the  three  following  : 
1.    u  z=f(x,  y,  z),    where    x  =  f,     y  —  f,z=-. 


»-- --i^     dv  _  c  ^du      ^  o^u  _1  du 

^t.  dt~^    dx  dy      f  dz- 


2.    u  =  log  (x^—  y-),    where    x  =  a  cos  t,    y  =  a  sin  t. 

—  =  -  2  tan  2  t 
dt 

3-    w=tan^^-,   where    x  =  2^,    ti  =  l  —  f.  —  =  - — 

y  dt      l  +  f. 


TARTIAL   DIFFERENTIATION'  143 

Apply  (10)  to  the  two  followiiii;: 

4.  u  =f(x,  II,  z),  where    /)  =  x-  -x,    z^x"-  af. 

^^  =  ^+(2..-!)^ +  (3ar'-2.)^. 
dx     dx  '  dii      ^  '  Qz 

5.  M  =  tan-i*^,  where    ?/  =  ,•] -a-s     z  =  \—?>x",  —  =  -J'. 

2;  dx      l+.r- 

Find  the  total  differential  by  (6)  or  (8)  in  the  followinj^ : 

6 .  u  =  ax-  +  2  bx>j  +  af,  du  =  2  (ax  +  hij)  dx  +  2  {hx  +  oj)  dy. 


f\o"  y  J     ,  lo"  a;  ,  ' 

u  f  — ^^  dx  -\ 5=—  dy 

y    X  y       \ 


7.    w  =  .^•'"^^  du 


0  ,      sin  \  (x  +  y)  ,        sin  v/  r/.P  —  sin  x  d>/ 

°  sin  i  (x  —  ?/)  cos  a-  —  cos  // 

>  9 .    ?f  =  ax-  +  />  i'/^  +  c;<;-  +  2  /?/^  +  2  5r;^.r  +  2  /<.r y,  ^ 

rti<  =2  («x  +  lvj+(jz)  dx  +  2(/^x-  +  by+fz)  dy  +  2((jx  -\-fy  +  C2)f/.t. 

.10.    ?t  =  x^",  du  =  .»;*'"'  (j/2;  dx  +  2;.«  log  .r  dy  +  a-^  log  .r  dz) . 

-^11.    7<  =  tan-  X  tan- y  tan- 2;,  rf«  =  4  u  { -^^—  +  -^  +  -^^. 

\sin2x-     sn\2y     sm27' 

If  the  variable  t  in  (5)  and  (7)  denotes  the  time,  we  have  the  n 
lation  between  the  rates  of  increase  of  the  variables.- 

For  illustration  consider  the  following  e:xanij)le : 

12.  One  side  of  a  plane  triangle  is  8  feet  long,  and  increasing  4 
inches  per  second;  another  side  is  5  feet,  and  decreasing  2  inches 
per  second.  The  included  angle  is  (50°,  and  increasing  2°  i»er  seconW 
At  what  rate  is  the  area  of  the  triangle  increasing  ? 


t4 

DIFFEREXTIAL   CAIXULUS 

/ 
The  arfea 

A  =  -  he  sin.  A,    from  which 

=  .4934  sq.  ft.  =  71.05  sq.  in.  per  sec. 

13.  One  side  of  a  rectangle  is  10  inches  long,  and  increasing  uni- 
formly 2  inches  per  second.  The  other  side  is  15  inches  long,  and 
decreasing  uniformly  1  inch  per  second.  At  what  rate  is  the  area 
increasing  ?  Ans.  20  sq.  in.  per  sec. 

At  what  rate  after  the  lapse  of  2  seconds  ? 

Ans.   12  sq.  in.  per  sec. 

14.  The  altitude  of  a  circular  cone  is  100  inches,  and  decreasing 
10  inches  per  second,  and  the  radius  of  the  base  is  50  inches  and 
increasing  5  inches  per  second.  At  what  rate  is  the  volume  in- 
creasing ?  Ans.   15.15  cu.  ft  per  sec. 

15.  In  Ex.  12,  at  what  rate  is  the  side  opposite  the  given  angle 
increasing  ?  Ans.   4.93  in.  per  sec. 

■  114.   Differentiation   of  an   Implicit  Function.    (See  Art.  66.)     The 

derivative  of  an   implicit  function  ma}^  be  expressed   in   terms  of 
partial  derivatives. 

The  equation  connecting?/  and  x,  by  transposing  all  the  terms  to 
one  member,  may  be  represented  by 

<^(^-,.v)=0 (1) 

Let  •  u==4>{x,y).     ,        -'^N.  J'':  ,'Xl  -  0 


From  (9),  Art.  113,  we  have  for  the  total  derivative  of  u, 

du  _dH      du  d>/ 
dx      dx      dy  dx 


PARTIAL    DIFFKUKNTIATLON  145 

But  by  (1)  X  and  v  mu&t  have  such  vahies  that  u  may  be  zero,  that 

'In 

dx 


is,  a  constant;  and  therefore  its  total  derivative—  must  be  zero 


Hence  f!i  +  ^^  =  0, 

dx      dy  dx 

du 

and      '  dy^_'d^ 

dx  du  ^  ^ 

dy 

For  example,  find  —^  from  afy-  +  x-y^  =  a*. 


Let  u  =  xb/  +  afy^  —  a*. 

4 

ax  dy 

•'^  ^"^  dx-  2x-3?/  +  3ary  2a;=^  +  3a-?/' 

In  the  same  way  find  the  first  derivatives  in  the  examples  of  Art.  C)fi. 


115.     Extension   of   Taylor's    Theorem   to    Functions  of  Two   Inde- 
pendent Variables.     If  we  apply  Taylor's  Theorem 

to  f(x+h,y-\-k), 

regarding  x  as  the  only  variable,  we  have 

^d_ 
'dx 


f(x+  h,  y  +  k)  =f(x,  y+k)  +  hf/{x,  y  +  k) 


+|^^/(,,, +  .)  +  -.  (1) 


146  DIFFERENTIAL   CALCULUS 

Now  expanding   f(x,  y  +  ^•);    regarding  y  as  the  only  variable, 
f{x,y  +  k)  =f{x,  y)  +  kj-^f(x,  y)  +,f  ^,/(^-,  V)  +  -• 


Substitnting  this  in  (1), 

fix  +  h,  y  +  A:)  =/(.r,  y)  +  h^  f(x,  y)  +  k^  f(x,  y) 
ox  dy 

This  may  be  expressed  in  the  symbolic  form  thus : 

f(x  +  h,y-\-  k)  =f{x,  y)  +  (h^+  k-f)  f(x,  y) 
\   dx         dyj 

where  f/i [-k — ]  is  to  be  expanded  by  the  Binomial  Theorem,  as 

\    dx        dyJ 

if    h—  and  k—  were  the  two  terms  of  the  binomial,  and  the  result- 
Si-  dy 

ing  terms  applied  separately  tof(x,  y). 

116.  Taylor's  Theorem  applied  to  Functions  of  Any  Number  of  In- 
dependent Variables.  By  a  method  similar  to  that  of  the  preceding 
article  we  shall  find 

f{^c  +  li,y  +  k,z^-l)=f{x,yr^)  +  (lil^  +  T^^  +  l£^f{x,y,z) 


This  expansion  may  be  extended  to  any  number  of  variables. 


PARTIAL    DIFFERENTIATION  147 

EXAM  PLES 

1.  Expand  log  (x  +  h)  log  (y  +  k). 

Let.  =/(..,  ,/)  =  log..log^,  |!^  =  120,    3«^logx 
da;         a;         oy         v 

d-n  __  _  logy         d-H    _  1       d^u  _       log  x 
dx-  X-    '     dydx      xy'     dy-  y- 

By  (2),  Art.  115,    log  {x^-li)  log  {y^k)  =  logx  logy 

,    /i  1  ,  ^"  1  li'   ,  ,    Ilk        k-   , 

+  -logy  +  -\ogx—  j—^  logy  H ^~  loga;+---. 

a-  y  Jx-  xy       2y- 

2.  (x  +  hy{y  +  kf  =  xY  +  3  hxY-  +  2  A'-r^ 

+  3  lih-y-  +  G  hkx-y  +  ^-V  +  •  •  •. 

3.  sin  [(ic  +  /<)  (y  +  k)']  =  sin  (.ry)  +  /iy  cos  {xy)  +  Au-  cos  (xy) 

—  -^sin  (.ry)  +  lik  [cos  (a-y)  —  .ry  sin  {xy)']  -  ~  sin  {xy)  +  •••• 


4.    log  (e^+*  +  6"+^)  =  log  (e^  +  e")  + 


he'  +  ke'      e'e'{h-kf 
e'  +  e'        2(e'  +  e»')- 


CHAPTER   XII 
CHANGE  OF  THE  VARIABLES   IN  DERIVATIVES 

fill     ci  11      fill  fl'V     fl~'V      (1  Oi* 

117.   To  express --^,  -4,  -y{,   •••  in  terms  of  -^,  --,  -r,,"'- 
dx   dx-    dx^  dij    dy    dif 

This  is  changing  the  uidepeiident  variable  from  x  to  y. 
By(l),Art.56,  1  =  1 (1) 


By  (3),  Art.  56, 
From  (1), 


Similarly, 
From  (2), 


dy 

d^y _d^dy_d^dy   dy 
dx?     dx  dx     dy  dx  dx 

d^ 

d  dy  _  d   1  _        dy^ 

dy  dx     dy  dx  ~ 

dy 

d^ 
dhj  _        dy- 

'''d^~~7d^' 

dhi  _  d  d?y_  _  d_  dry    dy^ 
dx^     dx  dx-      dy  dx-    dx 

/d-.i-V     dx  d^x 
d  d-y       \difj      dydif 


dydx^ 

\dy) 

^d-x\-     dx  d^x 


/da 
[di 


^dyy 
dhj     ^\d>rj       dy  dy^ 


dx"" 

148 


u. 


(2) 


CHANGE   OF   THE    VARIABLES    IN    DERIVATIVP:S       149 

It  is  sometimes  necessary  in  the  derivatives, 

dy    ^     0?y 

dx    dx^^    doi?''       ' 

to  introduce  a  new  variable  z  in  place  of  x  or  y,  z  being  a  given 
function  of  the  variable  removed. 

There  are  two  cases,  according  as  z  replaces  y  or  a*. 

■  118.    First.     To  express  ^'^  ^„  '^.  ...,  in  terms  of  ^  '^X  ^,  ..., 
dx  (/..-'  dx''  dx  dx^  dx^       ' 

where  y  is  a  given  function  of  z. 

By  (3),  Art.  56,      "  dy^<h,ch^ 

dx     dz  dx 

d-y  _  d /'dy\dz      dyd-z_d-yUlz'V    dy  ^ 
dx-     dx\dzjdx     dz  dot?      dz-\dx)     dz  dy?' 

Similarly,  we  find 

cfj[  _  rP^  /^^ Y I  o  f^  dz^  dh      dy  ^z_ 
da^      dz\dxj        dz^  dxd.rr      dz  dx^' 

Similarly,       -^,    --^,    .-.,  may  be  expressed  in  terms  of  z  and  x. 

It  is  to  be  noticed  that  in  this  case  there  is  no  change  of  the  " 
dependent  variable,  which  remains  x. 
For  example,  suppose    y  =  z\ 

Then  ^  =  3z'^. 

dx  dx 


q=ez(. 

dx"  V' 


dx  djy" 


iE  =  Q(^I£\'+lSz'^'^  +  3z^% 
dx^        \clxj  dxdx^  djf 


150  DIFFERENTIAL   CALCULUS 

119.     Second.     To  express^,  ^,  ^,  ..., 
dx   dx^    dx^ 

in  terms  of    -^ ,  — ^,  -^,  ....  where  x  is  a  given  function  of  z. 

dz   dz^    dz'^ 

This  is  chariging  the  independent  variable  from  x  to  z. 

dy 

By  (3),  Art.  56,  dy^dydz^dz^ 

^  ^  dx     dz  dx      dx 

■  dz 

cifdy\ 
d^y _  d  fdy\dz  _dz\dxj 
da?     dz\dxjdx         dx 
dz 

dx  d^y  dy  d^x 
_  dz  dz^  dz  dz^ 
~  7dx\^ 

Similarly,  higher  derivatives  may,  be  expressed.     In  practice  it 
is  generally  easier  to  work  out  each  case  by  itself. 
For  example,  suppose     x  =  z^. 


dy  _  dy  dz 
dx     dz  dx 

Buf 

dx      r,.    dz      z-^ 

d'z-'^^'Tx-'s- 

Hence 

d}/     1  ^-2  dy 

dx     3      dz' 

(1) 


d^^±(dy\±(dy\dz_ 
dx''     dx\dxj     dz\dxjdx 


CIIAXGK   OF   THE    VAUIAIILKS    IN    DKlil VATIVKS       VA 

Similarly,         ty^^dfOrjAdz^ 
d.v     dz\dxrjdx 

From  (2),  If^^  =  lf.-^/- 6.-^^^  10.-^/ 
^  ^  dz\dxy      9V      dz"  dz'  dz 

Hence  ^  =  1  f  .-^  _  6  .-^^  + 10  .-«^\ 

EXAMPLES 

Change  the  independent  variable  from  x  to  y  in  the  two  following 
equations : 

.  1.    3  f^y-  ^^-  '^fm^  0.  Am.   ^'  +  ^i!^  =  0. 

\djr  J      dxdx"      dx\dxj  di/      dy- 


c/x*        J\dx'J       \   dx       Jdx  dx 

Change  the  variable  from  y  to  z  in  the  two  following  equations: 
ax-  1+2/"  V"'*'/ 

Change  the  independent  variable  from  x  to  z  in   the   following 
equations : 

dcc^^ajda;     -^        '  dz'      dz 

o     <V^_2^fl/_^_y__,  =  0,     a-  =  tanz.        Ans.    y;  +  .V  =  " 


dar*     1  +  x-da;     (1  +  a^f 


dz' 


152  DIFFERENTIAL   CALCULUS 

^  dx^  dx 


d^  dz-        dz 


dx*  da?  dxr  dx 


s.    ^  +  2^  +  2/  = 
dz'        dz^     ^ 


yL20.  Transformation  of  Partial  Derivatives  from  Rectangular  to  Polar 
'Coordinates. 

Given  u  —  f(x,y), 

du        I    du    ■     .  r    dxi         -,    du        ■, 

to  exi)ress  —  aiicl  —   in  terms  oi   —    and   — ,    where  x,  ?/,  are  rec- 
dx  dy  dr  89 

tangular,  and  r,  '6,  polar  coordinates. 

We  have  from  (5),  Art.  113,  regarding  u  as  a  function  of  r  and  6, 

du  _  du  dr  .   du  dO  /-. s 

5^~a^5^    5^a^' ^  ^ 

d\i c)'i  dr      d}(  do  /on 

dy      dr  dy      dO  dy 

The  values  of    — ,  — ,   — ,  — ,  are  now  to  be  found  from  the  rela- 
dx    dy     dx    dy 

tions  between  x,  y,  and  r,  6. 

These  are  a;  =  rcos^,     ?/  =  ?-sin^.      .     ......     (3) 

But  in  the  partial  derivatives   — ,  — ,  and  — ,  — ,   r  and  6  are  re- 
ox*   dy  dx   dy 

garded  as  functions  of  x  and  y. 

These  are,  from  (3), 

r-^x'  +  y',     ^  =  tan-i^. 


CHANGE    OF   THH    VAIMAHLKS    IN    DKUIVATIVKS        153 
Differentiating,  we  find 

—  =-  =  cos(9,      --  =  ^  =  sin^, 
ax      r  dy      r 

—  =  _      ?/      _  _  sin^      89  _      X      _  cos  0 
dx  .«-  +  y-  r    '     dy      x-  -\-y-         r    ' 

Substituting  in  (1)  and  (2),  we  have 

^=cos^^-^^^, (4) 

dx  dr         r     dO  ^  ' 


du         .     a  8i(    ,    COS  6  du.  /-,, 

=  sin^  — + — (5) 

dy  dr         r      d9  •^ 


121     Transformation    of  — '-\ ;  from   Rectangular   to  Polar  Co- 

dx-      dy 

ordinates.    By  substituting  in  (4),  Art.  120,  —   for   u,  we  have 

d-u  _  8  /'du\_       ^  d  fdii\      sin  ^  5  f^"^  n\ 

d?~dx{dij~^^^    d^\dxj      We\dx) ^' 


Differentiating  (4),  Art.  120,  with  respect  to  r, 

8_fdn\_        ^d-u      sin  9  d'-n       sin  9  du  /r,>. 

drKdxj'^'^^    'd?        r    drdd        r"    86 ^"^ 

Differentiating  (4).  Art.  120,  with  respect  to  9, 

9  /3?A  ^  d-n         .     ^dii      s\n9d-n      cos  ^  5»/  ... 

■ — (  —  )=cos^  —  sin0^„ -—- ^„.     .     .     (•■ 

d9\dxj  drd9  dr        r     dff"         r     d6 


154  DIFFERENTIAL   CALCULUS 

Substituting  (2)  aud  (.T)  iu  (1),  we  have 

d~u  1  nd'u      2  sin  ^  cos  ^  d'u     ,  b\yi^  0  d'u  ,   p.m'^Odu 

—  =  cos  V H T- 

dx"  dr  r  drde  t"    dO'  r    dr 


2  sin  0  cos  0  du 


oe 


Similarly  by  using  (5),  Art.  120,  instead  of  (4),  we  find 

'^""  _  .•   2  A c)-«      2  sin  0  cos  0   d^a       cos^  0  dhi      cos^  6  du 
5/  ~  ''"^     d?  r  drdO         r-     86-         r      dr 

2  sin  0  cos  0  du 


de 


Adding  (4)  and  (5)^  -we  obtain 


d^u  ,   d-u  _  d'U  ,  1  du       1  d^ 
daf      dy^     di^      r  dr      i-  d6^' 


(4) 


.(5) 


CHAPTER  XITI 

MAXIMA  AND  MINIMA  OF  FUNCTIONS  OF  TWO  OR  MORE 
INDEPENDENT  VARIABLES 

122.  Definition.  A  function  of  two  indej)endent  variables, /'U-,//). 
is  said  to  have  a  niaxiiuum  value  wlien  x  =a,  y==b;  when,  for  all 
sufficiently  small  numerical  values  of  h  and  Jc, 

f(a,h)>f(a  +  h,b  +  k), (a) 

and  a  minimum  value,  when 

f(a,b)<f(a  +  h,b-{-Jc) (b) 

123.  Conditions  for  Maxima  or  Minima. 
If  u  =f(x,  y), 

we  find  that  a  necessaiy  condition  for  both  (a)  and  (6)  is  that 

—  =  0,   and    -=0,  when.T  =  a,    y  =  b. 
dx  dy 

This  may  be  shown  as  follows  : 

Conditions  (a)  and  (b)  must  hold  when  k  =  0,  and  we  have  for  a 
maximum 

f(.a,b)>f(a  +  h,b), 

i.  and  for  a  minimum 

f{ci,b)<f{<x^h,b), 

for  sufficiently  small  values  of  h. 

155 


156  DIFFERENTIAL   CALCULUS 

We  thus  have  for  consideration  a  function  of  only  one  variable. 
By  Art.  106,  we  must  have  for  both  maximum  and  minimum, 

—  f(x,  b)  =  0,  when  x  —  a, 
die' 

that  is  3~-^(^'  y)  ~  ^'  "^^6^  x  =  a,  y  =b. 

Similarly,  b}'-  letting  h  =  0  in  (a)  and  (b),  we  may  derive 
Q-  f(^,  y)  =  0,  when  x  =a,  y  =  b. 

These  conditions  for  a  maximum  or  minimum  are  necessary  but 
not  sufficient.  As  in  the  case  of  maxima  and  minima  of  functions 
of  one  variable,  there  are  additional  conditions  involving  derivatives 
of  higher  orders.  These  we  shall  give  without  proof,  as  their 
rigorous  derivation  is  beyond  the  scope  of  this  book. 

The  conditions  for  a  maximum  or  minjmum  value  of  u  =  /(x,  y) 
are  as  follows : 

For  either  a  maximum  or  minimum, 

!-:=»'  -^  !-:='■' w 

\dy  dxj       dx-  dy^ 

For  a  maximum,       — !<0,     and      —  <0 (3) 

dx-  dy- 

For  a  minimum,        — !>0,     and       — '  >  0 (4) 

dx'  dy-  ^  ^ 

124    Functions  of  Three   Independent   Variables.     The   conditions 
for  a  maximum  or  minimum  value  of  u=f(x,  y,  z)  are  as  follows: 
For  either  a  maximum  or  minimum, 

du  __  0   du  _r.    dii  _  ,-. 
dx        '  dy        '  dz 

and  /_Sh^y^d'„Sh. 


\dx  dyj 


dx-  dy'^ 


MAXIMA   AND   MINIMA   OF   FUNCTIONS 

For  a  maximum  — '^  <  0,  aud  A  <  0; 

5  a;- 


t'or  a  minimum, 


d-ii 


>0,  aud  A>0: 


where  A 


d^u  d-u  d-u 
dxr     dx  dy     dx  dz 

d-u  dhc  dhi 
dx  dy    dy^'    dy  dz 

d^u  dhi  d-u 
dx  dz'    dy  dz'    dz'^ 


EXAM  PLES 
1.    Find  the  maximum  value  of 

u  =  3  axy  —  x^  —  y^. 


Here 


dU  r,  r,       ->  du  r,  no 

—  =  3ay-3x-,    —  =  3ax-3y\ 
dx  dy 


d3T  ay-  dx  dy 


Applying  (1),  Art.  123,  we  have 

ay  —  X-  =  0,     and     ax  —  y-  =  0 ; 


whence 


a;  =  0,  ?/  =  0;  or  a;  =  a,  //  =  a. 


The  values  x  =  0,y  =  0,  give 


^^  =  0,  ^  =  0,  -^  =  3a, 
dx?  dy-  dx  dy 


which  do  not  satisfy  (2),  Art.  123. 

Hence  they  do  not. give  a  maxjinum  or  minimum. 


158  DIFFERENTIAL   CALCULUS 

The  values  x  =  a,  y  =  a,  give 

■7-0  =  — Q  a,   -^  =  —  6  a,    -——  =  3  a, 
oar  dy-  dxdy 

wliich  satisfy  both  (2)  and  (3),  Art.  12.3. 

Hence  they  give  a  maximum  value  of  u,  which  is  a'. 

2.    Find  the  inaximuin  value  of  xyz,  subject  to  the  condition 

<  +  ;C  +  ^  =  l. (1) 

a-     h'     c- 

ri'om(l)  -!  =  l--,-f^,; 

c-  a-      b- 

and   as  xyz  is  numerically  a  maximum  when  x^y-z^  is  a  maximum, 
we  put 

o    2  A  ^■'         V' 


5a;  5i/ 

From  —  =  0  and  —  =  0,  we  find,  as  the  only  values  satisfying 
dx  dy  ^  ^     ^ 


(2),  Art.  123, 


X  = ,    y  = ,    which  give 

V3'    ^      V3' 

dx"  9  '    dy'  9  '    6a;ay  9  ' 


MAXIMA   AND   MINIMA    OF   FUNCTIONS  loO 

As  these  values  satisfy  (2)  and  (3),  Art.  323,  it  follows  that  xyz  is 
a  maxiiiuuu  when 

V3  V3  V3 

The  maxinnim  vahie  of  xyz  is  ""  ^  • 
'        3V3 


3.    Find  tlie  values  of  x,  y,  z  that  rend 


er 


s^  +  y-  +  z-  +  x  —  2z  —  xy 

a  minimum.  .  2  1  ^ 

Ans.    x=--,    2/=  --,    2  =  1. 

I    4.    Find  the  maximum  value  of 

(a  —  x)(a  —  y)(x -{- y  —  a).  Ans.   ~. 

27 

/     5.    Find  the  minimum  value  of 

x'  +  xy  +  y-  —  ax—  by.     Ans.  -  {ah  —  a-  —  b'^). 
o 

6.    Find  the  values  of  x  and  y  that  render 

sin  X  +  sin  y  +  cos  (.x  +  y) 

a  maximum  or  minimum.  -<  a       •    •  i  ■^ 

Ans.     A  minimum,  when  x  =  y  =  -j~\ 


a  maximum,  Avheu  x  =  >/  =  -,. 


7.    Find  the  maximum  value  of 


(ax  +  by  +  cy  Ans.a'^  +  b"-  +  c^. 

^"^  +  ^  +  1 


8.    Find  the  maximum  value  of  x^tfh*.  subject  to  the  condition 


2x  +  3y  +  4z  =  a  Ans.     [^ 


9.    Find  the  minimum  value  of  -  +  -  +  ^,  subject  to  the  conditi.  ■ 

a      b      c 

xyz  —  abc.  Ans.    3 


160  .  DIFFERENTIAL   CALCULUS 

10.  Divide  a  into  three  parts  such  that  their  continued  product 
may  be  the  greatest  possible. 

Let  the  parts  be   x,  y,  and  a  —  x  —  y. 

Then  w  —  xy{a  —  x  —  y),  to  be  a  maximum. 

—  =ay-2xy-y-^Q,    —  =  ax-x'-2xy  =  0. 
dx  dy 

These  equations  give  x  =  y  —  -. 

Hence  a  is  divided  into  equal  parts. 

Note.  —  When,  from  the  nature  of  the  problem,  it  is  evident  that  there  is 
a  niaxiriium  or  minimum,  it  is  often  unnecessary  to  consider  the  second 
derivatives. 

11.  Divide  a  into  three  parts,  x,  y,  %,  such  that  x'"y'^z^  may  be  a 
maximum. 

X      y     z  a 


Ans. 


m      n     p      m  +  n  -\-p 


•I  12.  Divide  30  into  four  parts  such  that  the  continued  product  of 
the  first,  the  square  of  the  second,  the  cube  of  the  third,  and  the 
fourth  power  of  the  fourth,  may  be  a  maximum. 

Ans.   3,  6,  9,  12. 

13.    Given  the  volume  a^  of  a  rectangular   parallelopiped ;    find 
when  the  surface  is  a  minimum. 

Ans.   When  the  parallelopiped  is  a  cube. 

V  14.  An  open  vessel  is  to  be  constructed  in  the  form  of  a  rec- 
tangular parallelopiped,  capable  of  containing  108  cubic  inches  of 
water.  What  must  be  its  dimensions  to  require  the  least  material  in 
construction  ? 

Ans.    Length  and  widtli,  6  in.  ;   height,  3  in. 

15.    Find  the  coordinates  of  a  point,  the  sum  of  the  squares  of 
whose  distances  from  tliree  given  points. 


MAXIMA    AND    MINIMA    OF    FINCTIOXS  llil 

is  a  miuinuun.  ^         1  /  ,1 

Ans.   -  (.r,  +  -r,  +  .fa),    -  (//.  +  >h  +  ^3), 

the  centre  of  gravity  of  the  triangle  joining  the  given  points. 

16.    If  X,  y,  z  are  the  perpendiculars  from  any  point  P  on  the  sides 
(_^    a,  6,  c  of  a  triangle  of  area  A,  tind  the  niiniinuiu  value  of  .r  +  y-  +  z-. 

4A- 


Ans. 


17.   Find  the  volume   of  the   greatest  rectangular  parallelepiped 
that  can  be  inscribed  in  the  ellipsoid, 

^  +  ^  +  ^  =  1.  Ans.  ^^'. 

a-     b-     c-      .  3V;3 


18.    The  electric  time  constant  of  a  cylindrical  coil  of  wire  is 

mxyz 

u  = , 

ax  +  by  +  cz 

where  x  is  the  mean  radius,  y  is  the  difference  between  the  internal 
and  external  radii,  z  is  the  axial  length,  and  m,  a,  b,  c  are  known  con- 
stants. The  volume  of  the  coil  is  nxyz  =  g.  Find  the  values  of  x,  y,  z 
which  make  u  a  minimum  if  the  volume  of  the  coil  is  fixed  ;  also  the 
minimum  value  of  u. 


Ans.   ax=^by^cz=J\'^.    u  =  '^\\-t-. 


CHAPTER   XIV 


CURVES  FOR  REFERENCE  ^ 

We  give  in  this  chapter  representations  and  descriptions  of  some       ■ 
of  the  curves  used  as  examples  in  the  following  chapters. 


RECTANGULAR  COORDINATES 
Y 


125.     The  Cissoid,        ^St^ihJ'e 


^'      2a- X 

This  curve  may  be  constructed  from 
the  circle  ORA  (radius  a)  by  drawing 
any  oblique  line  OM,  and  making 

P3I=  OR. 

The  equation  above  may  be  easily 
obtained  from  this  construction.  The 
line  ^iW  parallel  to  OY  is  an  asymp- 
tote. 

The  polar  equation  of  the  cissoid  is 

r  =2a  sin  ^  tan^. 


162 


C|L)UVES  FOR   REFKRKNCE 


126.    The  Witch  of  Agnesi,  ij  = 


This  curve  may  be  constructed  from  the  circle  ORA  (radius,  a)  by 
drawing  any  abscissa  3IR,  and  extending  it  to  P  determined  by  ORN, 
by  the  construction  shown  in  the  figure. 

The  equation  above  may  be  derived  from  this  construction.  The 
axis  of  X  is  an  asymptote. 


127.    The  Folium  of  Descartes, 

of  ^f-2,axy^  0. 

The  point  A,  the  vertex  of 
the  loop,  is 


/3a  3a\ 


The  equation  of  the  asymp- 
tote MN  is 

x-\-y-\-a=0. 

The   polar   equation    of    the 
folium  is 

,  ^  3  rttan  e  sece 
''        l  +  tan«^ 


a  0 
a 


164 


DIFFERENTIAL   CALCULUS 


128 .     The  Catenary,       ?/  =  ^  ( e«  +  e  «) . 


This  is  the  curve  of  a  cord  or  chain  suspended  freely  between  two 
points. 

129.     The  Parabola,  referred  to  Tangents  at  the  Extremities  of  the 
Latus  Rectum,  x'^  -\-  y^  =  ct'^- 

OL  =  OL'  =  a. 

Y 


The  line  LL'  is  the  latus  rectum ;  its  middle  point  F,  the  focus ; 
0F3f,  the  axis  of  the  parabola ;  A  the  middle  point  of  OF,  the  vertex.  • 


CURVES   FOR   KKFKUKN'CE 


!(]: 


130.     The  curve  a"  \y  =  a*",  where  one  coordinate  is  proportional 

to  the  ?ith  power  of  the  other,  is  sometimes  calk'd  tlie  jHirabola  of 
the  nth  degree. 

If  n  =  3,  we  have  the  Cubical  Parabola,  a-ij  =  x\ 


lin=f,  we  have  the  Semknbical  Parabola, 
a'^y—  X',  ay-  =  x^. 


ine  DIFFERENTIAL  'CALCULUS 

131.    The  Two-arched  Epicycloid. 


x-=  -^cos<^ cos  3^, 


3a  .     ,      a  •    o  , 
y=  — -SHK^  --sm3<^. 


132=     The  Hypocycloid  of  Four  Cusps  sometimes  called  the  Astroid, 


x^  +2/3  =  a'i- 

This  is  the  curve  de- 
scribed by  a  point  P 
in  the  circumference  of 
the  circle  PR,  as  it  rolls 
within  the  circumfer- 
ence of  the  fixed  circle 
ABA',  whose  radius  a 
is  four  times  that  of 
the  former. 

The  equation  above 
may  be  given  in  the 
form 

<r=acos^()!),  y=asii\^<ji. 


CURVES  •von    UKFKKENCE 


133.  The  Curve, 

The  equation  is 
the  same  as  that 
of  the  ellipse  with 
the  exponent  of 
the  second  term 
ehanged  from  2 
to  |. 

134.  The  Curve,  a' 


(«)'- 


f--> 

'  =  1. 

Y 

w 

B 

/. 

"X 

-iu 

,^ 

n     ^--~. 

...fl 

v^ 

0                   ^^^ 

POLAR    COOIIDLXATES 
135.    The  Circle,  r  =  a  §in  ^  +  &  cos  B. 


By  laying  off  OB  =  />,  and  JIA  =  a,  we 
determine  OA  the  diameter  =  Va'^  +  b'. 

If  6  =  0,  r  =  a  sin  6,  the  circle  referred- 
to  OX.     (2d  fig.) 

If  a  —  0,  '/•  =  b  cos  0,  the  circle  referred 
to  O'X'. 


168  DIFFERENTIAL  CALCULUS 

136.    The  Spiral  of  Archimedes,  r  =  a( 


In  this 
curve  r  is 
proportional 
to  6.  Lay- 
ing off 


OA, 


when 


then 


OPi  =  i-0^,    0^2  =  10^,    OPs  =  iOA,    OP,  =  ^OA, 
0B  =  2  0A,    OC  =  SOA. 
The  dotted  portion  corresponds  to  negative  values  of  6. 

137.  The  H3rperbolic  or  Reciprocal  Spiral,  rd  —  a. 

In  this  curve  r 
varies  inversely  as 
e.  The  line  3fN 
is  an  asymptote, 
which  the  curve 
approaches,  as  $ 
approaches  zero. 

Since  r=0  only 
when  ^  =  Go  ,  it  fol- 
lows that  an  in- 
finite number  of 
revolutions  are 
necessary  to  reach  the  origin. 


CURVES    FOR    RKFHRENCE 


109 


138.    The  Logarithmic  Spiral 

Starting  from  ^1, 
where  ^=0  and  r=l, 
r  increases  with  6 : 
but  if  we  suiDpose  $ 
negative,  r  decreases 
as  0  numerically  in- 
creases. Since  r=0 
only  when  0—~cx), 
it  follows  that  an 
infinite  number  of 
retrograde  revolu- 
tions from  A  is  re- 
quired to  reach  the 
origin  0. 

A  property  of  this  spiral  is  that  the  radii  vectores  OP,  OP^,  OP, 
make  a  constant  angle  with  the  curve. 


139.  The  Parabola,   Origin  at  Focus, 


cos  e)^2  a. 


The  initial  line  OX  is  the  axis  of 
the  parabola;  the  origin  0  is  the 
focus ;   LL',  the  latus  rectum. 


140.   The  Parabola,  Origin  at  Vertex  (see  preceding  figure), 

T  sin  6  tan  6  =  4a. 
The  initial  line  is  the  axis  AX;  the  origin  is  the  vertex  J 


170 


DIFFERENTIAL   CALCULUS 


141.   The  Cardioid,   r  =  a(l  —  cos  0). 

This  is  the  curve  described 
by  a  point  P  in  the  circiun- 
ference  of  a  circle  l^A  (di- 
ameter, a)  as  it  rolls  upon 
an  equal  fixed  circle  OA. 

Or  it  may  be  constructed 
by  drawing  through  0,  any 
line  OR  in  the  circle  OA, 
and  ])roducing  OB  to  Q  and 
Q',  making  EQ=EQ'=OA. 

The  given  equation  fol- 
lows directly  from  this  con- 
struction. 


142    The  Equilateral  Hyperbola,    r-  cos  2  (9  =  al 


The  origin 
0  is  the 
centre  of  the 
hyperbola, 
and  the  in- 
itial line  OX 
is  the  trans- 
verse axis. 

If  or  IS 
taken  as  the 
initial  line, 
the  equation 
of  the  hyper- 
bola is 

?'-  sin  2  6=0.-. 


CURVES   FOR   REFERKNCE  171 

143.    The  Lemniscate  referred  to  0.1  (see  preceding  figure), 

r-  =  a-  cos  2  6. 

This  is  a  curve  of  two  loops  like  the  figure  eight. 

It  may  be  defined  in  connection  with  the  equilateral  hyperbola,  as 
the  locus  of  P,  the  foot  of  a  perpendicular  from  O  on  I'(^,  any 
tangent  to  the  hyperbola. 

The  loops  are  limited  by  the  asymptotes  of  the  hyperbola,  making 


TOX=rOX=Al 


OA  =  a. 


The  lemniscate  has  the  following  property : 

If  two  points,  F  and  F',  called  the  foci,  be  taken  on  the  axis,  such 


that 


0F=  0F'  = 


V2 


then  the  product  of  the  distances  P'F,  P'F',  of  any  point  of  the 
curve  from  these  fixed  points,  is  constant,  and  equal  to  the  square 
of  OF. 

If  02"  is  taken  as  the  initial  line,  the  etiuation  of  the  lemniscate  is 


144.    The  Four-leaved  Rose,    r  =  a  sin 
Y 


172 


DIFFERENTIAL   CALCULUS 


145.    The  Curve,    r  —  a  sin 


.6 


CHAPTER   XV 


DIRECTION  OF  CURVES.     TANGENTS   AND   NORMALS 


We  have  seen  in  Art.  17  that  the  derivative  at  any  point  of  a 
plane  curve  is  the  slope  of  the  curve  at  that  point.  We  will  now  con- 
sider some  further  applications  of  differentiation  to  curves. 

146.  Subtangent,  Subnormal,  Intercepts  of  Tangent.  —  Let  PT  be 
the  tangent,  and  FN  the  normal,  to  a  curve  at  the  point  P,  whose 
ordinate  is  y  =  PM. 
Then  MT  is  called  the 
subtangent,  and  MN 
the  siibnormcd,  corre- 
sponding to  the  point 
P. 

To  find  expressions 
for  these  cpiantities : 

Let  4>  tlonote  the 
angle  PTX,  the  in- 
clination of  the  tan- 
gent to  OX. 

By  Art.  17, 


Y 

F 

■>/ 

0 

0\ 

M               N 

T' 


tan  7^7'X 


Subtangent  =  TM=  PM  cot  PTM . 


If  ilx 

,1  cot  <i  =  ^  =  //       • 

dy        dy 


dx 


Subnormal  =  MN=  PM  tan  MPX=  y  tan  «^  =  y 


Intercept  of  tangent  on  0X=  OT^  OM 

Intercept  of  tangent  on  0  Y=  OT'  =  PS 
But  as  OT'  is  negative,  we  have 
Intercept  of  tangent  on  01'=  y  —  x  tan  <^  = 
173 


dx 


TM=x 
PM=  X  tan  4, 
(hi 

'  — X— ■• 

dx 


174 


DIFFERENTIAL   CALCULUS 


147.   Angle  6f  Intersection  of  Two  Curves.     Suppose  the  two  curves 
intersect  at  P. 

Let  PT  and  PT'  be  the 
tangents  at  P. 

PTX=<f>,    PT'X:=C}>', 

and    let    I    be    the    angle 

7'PT'  between  the  tangents. 

Then  /==<^'  — </>  and 


tan  /= 


tan  (^'  — tan  <^ 
1  +  tan  (f)'  tan  </> 


(1) 


From  the  equations  of 
the  given  curves  find  the 
coordinates  of  the  point  of 
intersection  P;  then  using 

these  equations  separately,  find  by   tan  ^  = '^   the  values  of  tan  <^ 

dx 
and  tan  4>'  for  the  point  P.     Substituting  in  (1)  gives  tan  /. 
For  example  find  the  angle  at  which  the  circle 


x^-^y-  =  13,      .     .     . 
intersects  the  parabola 
2?/-' =  9.1'.     .     .     . 


(2) 


(3) 


The  intersection  P  of  (2)  and  (3)  is 
found  to  be  (2,  3). 

Differentiating  (2), 


dl[ 
dx 


for  P,  tan  <i  = 


From  (3),  ^'-^1      ^' 

Substituting  in  (1),  tan  / 


dx     Ay      4 
17 


-  for  P, 


tan  </)' : 


/=70'33' 


DIRECTK^X    OF   CLUVES.      TANCxKNlS    A  M  >    XolLMAI.S      175 

EXAMPLES 
r  Find  the  direction  at  the  origin  of  the  curve, 

(a*  -  h*)  y^x(x~ay-  b*x.        Ans.  45°  witli  OX. 

What  must  be  the  relation  between  o  and  b,  so  that  it  may  be 

parallel  to  OX  at  the  point  a;  =  2  a  ?  Ans.     3  a-  —  Jr. 

2.    Find  the  points  of  contact  of  the  two  tangents  to  the  curve, 
6  y  =  2  x"  +  9  a--  -  1 2  x  +  2,  ' 
parallel  to  the  tangent  at  the  origin  to 
the  curve,  ?/  +  ay  ^  2  ax.  Ann.   A ,  1\  ( -  4, 1  !)• 

^    Z.   Find  the  subtangents  and  subnormals  in  the  parabolas, 
y-  =  A  ax,  and  ay^  =  4  ay. 
Ans.     Subtangents,  2.1;,  [^;  subnormals,  2  a, — -• 

'  4.    Find  the  aubtangent  and  subnormal  in  the  cissoid  (Art.  125), 
y^  =  —^ ,  at  the  point  (a,  a).         Ans.     -,    2  a. 

^b.    Show  that  the  sum  of  the  intercepts  of  the  tangent  to  the 

1  X  1 

parabola  (Art.  129),       x^-j-  y^=  a'-,  is  equal  to  a. 


6.    Show  that  the  area  of  the  triangle  intercepted  from   the  co 
gent  to  the  hyperbola, 

2  xy  =  a^,  is  equal  to  a-. 


ordinate  axes  by  the  tangent  to  the  hyperbola. 


7.  Show  that  the  part  of  the  tangent  to  the  hypocycloid  (Art.  lo2V 
x^  _|_2/*=aS     intercepted  between  the  coordinate  axes,  is  equal  to  a. 

8.  At  what  angle  do  the  jnirabolas,  y-  =  ax  and  sr  =  H  ay  iuti-rseft 

Ans.    At  (0,  0),  90";  at  another  point,  t:ur'  '.'• 


176  DIFFEREXTrAL   CALCULUS 

9.    At  what  angle  does  the  circle,  x-  -\-  y-  =  5  x,  intersect  the  curve, 
8y  —  7x^  —  l,  at  their  common  point  (1,  2)  ?  Ans.     45°. 

10.    Show  that  the  ellipse  and  hyperbola, 

^"  +  ■1'  =  !      ^"_?/!  =  l 
7      2        '3       2        ' 

intersect  at  right  angles. 


11.    Find  the  angle  of  intersection  of  the  circles, 

o(r-\-y-  —  x  +  3y  +  2  =  0,      x-  +  y-  —  2y  =  9.         Ans.     tan"^  -. 


^12.    Show  that  the  parabola  and  ellipse, 

y-  =  ax,     2  xr  +  ?/-  =  b^, 
intersect  at  right  angles. 

^^  13.    Show  that  the  parabolas, 

2/-  =  2  ax  +  tt'-,  and  x^  =  2by  +  b^, 
intersect  at  an  angle  of  45°. 

/14.    Find  the  angle  of  intersection  of  the  parabola, 
a-  =  4  ay,  and  the  witch  (Art.  126),  y         ^^' 


x'  +  Aa^ 
A71S.     tan"^  3  =  71°  34'. 

15.    Find  the  angle  of  intersection  between  the  parabola, 
?/-  =  4  ax,  and  its  evolute,  27  a?/^  =  4:  (x  —  2  af.    (See  Fig.,  Art.  167.) 

Ans.    tan~^ 

V2 

148.  Equations  of  the  Tangent  and  Normal.  Having  given  the 
equation  of  a  curve  ?/=  /'(.^■),  let  it  be  required  to  find  the  equation 
of  a  straight  line  tangent  to  it  at  a  given  point. 


DIRECTION   OF   CURVES.     TANGENTS   AM)    NOUMAJ.S      177 

Let  (x',  ?/')  be  the  given  point  of  i-ontact.     Tlien  the  ec [nation  of  a 
straight  line  through  tliis  point  is 

y-j/'  =  m{x-x'), (1) 

in  which  x  and  //  are  the  variable  coordinates  of  any  point  in  the 
straight  line;  and  m,  the  tangent  of  its  inclination  to  the  axis  of  A'. 
But  since  the  line  is  to  be  tangent  to  the  given  curve,  we  must  have, 
by  Art.  17, 

m  =  tan  c^  =  ^-', 
dx 

-^  being  derived  from  the  equation  of  the  given  curve  y  =/(.r), 

and  applied  to  the  point  of  contact  (x',  //'). 

If   we   denote    this    by  — ^,    we    have,    substituting   ??«.  =  ^^  in 
equation  (1),  '^•'■'  '  ^•^' 

y-y'  =  f^,ix-x'), (2) 

for  the  equation  of  the  required  tangent. 

Since  the  normal  is  a  line  through  (x',  //')  perpendicuhir  to  the 
tangent,  we  have  for  its  equation 

W— W= (x  —  X)= (X  —  X)'       ....  i) 

^      -^  dl^  ^  dy'^  ^  ^  ' 

dx' 

For  example,  find  the  equations  of  the  tangent  and  normal  to  the 
circle  x^  +  y'^  =  a-,   at  the  point  (x',  ?/'). 

Here,  by  differentiating   x-  +  y-  =  cr,  we  find 

^  =  -^,  from  which  ^-^'  =  -^' 
dx         y  dx  y' 


Substituting  in  (2),  we  have 


X 


,'  =  -'^(x-x% 

y 


as  the  equation  of  the  required  tangent. 


IY8  DIFFERENTIAL   CALCULUS 

It  may  be  simplified  as  follows: 

ra' +  2/?/'=  ^' '  +  ?/''  =  "'• 
The  equation  of  the  normal  to  the  circle  is  found  from  (3)  to  be 


which  reduces  to  -^y  —  y^- 

^  EXA^^PLES 

Find  the  equations  of  the  tangent  and  normal  to  each  of  the  three 
following  curves  at  the  point  {x\  y') : 

•    1.    The  parabola,    y-=z^ax. 

Am.   yy'=^2a[^x  +  x'),  2a{y -y') +y\x- x')  =0. 


y    2.    Theellip^e,  ^^  +  '^3  =  1. 

Ans.  -H^  +  ?!  =  1,  ^'^'(y  -  y')=o:hj\x  -  xy 

a-        0- 


/   3.    The  equilateral  hyperbola,   2  xy  =  cr. 

Ans.    xy'  +  yx'  =  cr,  y'{y-y')  =  x'(^x  -  x'). 

.   4.    Find  the  equation   of    the  tangent  at  the  point  (:x',y')  to  the 
ellipse,   3 ir^ -  4 xy  +  2  f- -\-2x  =  2. 

Ans.   3xx'  +  2yy'-2ix'y  +  y'x)+x  +  x'  =  2. 


5.    Find 


the  equations  of  tangent  and  normal  at  the  point  (x',  y') 


Ans. 


to  the  curve,  xf  =  aY 

x'       y' 


DIRECTIOX   OF   CURVKS.     TANGENTS   AND    NORMALS     170 

6.  In  the  cissoid  (Art.  125),  f  =  -—-,  fin.l  the  equations  of  the 

-  <i.  —  X 
tangent  and  normal  at  the  points  whose  abscissa  is  a. 

Ans.   At  (a,  a),        y  =  2  x  -a,     2y  +  x  =  'S  a. 
At(«,-a),     7/  +  2x  =  a,     2  7j=:x-So. 

7.  In  the  witch  (Art.  126),  y  =      ^/''^        find    the   equations    of 

4  a-  +  x"'    . 
the  tangent  and  normal  at  the  point  whose  abscissa  is  2  a. 

Ans.   x  +  2y  =  4:a,    »/  =  2.c-oa. 

8.  Find  tlie  equation  of  the  tangent  at  the  point  (x',  y')  to  the 
curve,   af y  +  xy-  =a'\ 

Ans.     xy'(2x'+7j')+y.v'(2y'  +  x')  =  :^a\ 

Find  the  equations  of  tangent  and  normal  to  the  three  following 
gujvep : 

1^9/  x'+f  =3  axy  (Art.  127),  at  fe  ^\      Ans.    x  +  y  =  -M,    .v^,. 

i^'  10.    x  +  y  =  2  e^-\  at  (1, 1).  Ans.    3 y  =  x  +  2,    3 .r  +  //  =  4. 

^  11.    /"-Y"  +  U\=  2,  at  (a,  h).     Ans.  -  +  ^  =  2,    ax  -  by  =  a-  -  // . 

■  12.    Find  the  equations  of  the  two  tangents  to  the  circle, 
.^■-  4-  y-  —  3 .?/  =  14,  parallel  to  the  line,  7  y  =  4:  x  +  1. 

Ans.   7  y  =  4:  X  -\-  4P>,    7  y  =  4x  —  22. 

i'^'   ) 

'\  15;   Find  the  equations  of  the  two  normals  to  the  hyperbola, 

4  a;2  _  9  ?/-  +  36  =  0,  parallel  to  the  line,  2  r/  +  5  x-  =  0. 

Ans.    8  y  +  20  .r  =  ±  Ga. 

149.  Asymptotes.*  When  the  tangent  to  a  curve  approaches  a 
limiting  position,  as  the  distance  of  the  point  of  contact  from  the 
t)rigin  is  indefinitely  increased,  this  limiting  position  is  called  an 

*  The  limits  of  this  work  allow  only  a  brief  notice  of  this  subject. 


180  DIFFERENTIAL   CALCULUS 

asymptote.  In  other  words,  an  asymptote  is  a  tangent  which  passes 
within  a  finite  distance  of  tlie  origin,  although  its  point  of  contact  is 
at  an  infinite  distance. 

We  have  found  in  Art.  146,  for  the  intercepts  of  the  tangent  on  the 
coordinate  axes, 

Intercept,  on  OX  =  a;  —  ?/  — ,     Intercept  on  0  Y=  y  ~x^-^. 
cly  dx 

If  either  of  these  intercepts  is  finite  for  x  =  co,  or  ?/  =  co,  the  cor- 
responding tangent  will  be  an  asymptote. 

The  equation  of  this  asymptote  may  be  obtained  from   its  two 

intercepts,  or  from  one  intercept  and  the  limiting  value  of  -^. 

dx 
Let  us  investigate  the  conic  sections  with  reference  to  asymptotes. 

(1)  The  parabola,  ?/-  =  4  ax,      -^  =  :=:^ . 
^  '  dx       y 

Intercept  on  OX  ::=x~y—  —  x—  f—  =  —  x, 

-r     .  .  r\-ir  dv  2  QX         V 

Intercept  on  OY  =  y  —  x—^  =  y =  ^. 

(7.*;  y        2 

When  a;  =  co,  ?/=  co,  and  both  intercepts  are  also  infinite. 
Hence  the  parabola  has  no  asymptote. 

(2)  The   hyperbola,  -^'-^  =  1,     ^1/  =  ^. 
^  ^  •'^       .         o?      h-  dx      a^y 

Intercept  on  OX  =  — ,     Intercept  on  0Y= . 

x  y 

These  intercepts  are  both  zero  when  a;  =  00,  and  there  is  an 
asymptote  passing  through  the  origin.  To  find  its  equation,  it  is 
necessary  to  find  the  limiting  value  of  — ,  when  x  =  co. 


dy  _  h-x  _             hx 

—  ±'i    ^ 

dx     d-y         a^^~ 

-"    "V-f 

Hence 

dx          a 

when  a;  =00. 

DIRECTION    OF   CUllVES.      TANGENTS    AND    NORMALS      ISl 
There  are  then  two  asymptotes,  whose  equations  are 

a 

(3)      The    ellipse,    having    no    infinite   branches,    can    liave    no 
asymptote, 

150.     Asymptotes  Parallel  to  the  Coordinate  Axes.     When,  in  the 

equation  of  the  curve,  x  —  co  gives  a  finite  value  of  //,  as  y  =  n,  then 
?/=  a  is  the  equation  of  an  asymptote  parallel  to  OX. 

And  when  y  =  cc  gives  x  =  a,  then  ic  =  a  is  an  asymptote  parallel 
to  OT. 


151.  Asymptotes  by  Expansion.  Frequently  an  asymptote  may 
be  determined  by  solving  tlie  equation  of  the  curve  for  x  or  y,  and 
expanding  the  second  member. 

For  example,  to  find  the  asymptotes  of  the  hyperbola 

a-     Ir 

As    X    increases    indefinitely,   the    curve    approaches   the    lines 

y  =  ±  -^ ,   the  asymptotes. 
a 


EXAMPLES 
Investigate  the  following  curves  with  reference  to  asymptotes : 

1     y  — — Asymptote,  y  =  x. 

^      x'+Sar' 

2.  f  =  ex-- :>?.  Asymptote,  x-\-y  =  2. 

3.  The  cissoid  (Art.  125)  f-  =  .^  ^  Asymptote,  x  =  2a. 


182  iJlFFEHEXTIAL   CALCULUS 

J  4.    X'  +  y"  =  a".  Asymptote,  a;  +  ?/  =  0. 

'■■'    5.    {x  —  2  a)//-  =  a;^  —  a'l  Asymptotes,  a;  =  2  a,  x-\-a—±  y. 

6.    x''  -I-  /'  =  3  axij  (Art.  127).  Asymptote,  x  +  ?/  +  a  =  0. 

(Substitute  y  —  vx  in  the  given  equation  and  in  tlie  expressions 
for  the  intercepts.) 

152-   Direction  of  Curve.     Polar  Coordinates. 

In  this  case  the  angle 
OPT  between  the  tangent 
and  the  radius  vector  may 
be  most  readily  obtained. 
Denote  this  angle  by  ij/. 
Let  r,  0,  be  the  coordi- 
nates of  P;  r  +  Ar,e  +  ^e, 
the  coordinates  of  Q. 
Draw  PE  perpendicular 
to  OQ. 


Then    tan  PQi^  =  ^ 


0 

T 

X 

r 

sin  AO 

r 

sin 

A^ 

r  +  A> 

-/•cosA^       ^, 

+ 

2'/ 

sin- 

A^ 
2 

sin  A^ 
A^ 

A^^ 

.     A^ 
r  sin  — 

.     A^ 

sm- 

A^    ' 

Now  let  A^  approach  zero ;   the  point  Q  approaches   P,  and  the 
angle  PQE  approaches  its  limit  ij/. 

Hence  tan  i// =  Lim^^^^otan  PQ72  =  --; (1) 

dO 
The  inclination  <^  of  the  tangent  to  OX  may  be  found  by 

<l>  =  ip  +  e (2) 


DIRECTION   OF   CURVES.     TANGENTS   AND    NORMALS     IS'.) 
153.  Polar  Subtangent  and  Subnormal. 


If  through  0,  NT  be  drawn  per-      N 
pendicular  to  OP,  OT  is  called  the 
2wlar  subUingeut,  and  ON  the  jwlar 
Hiihnormcd,     corresponding     to     the 
point  P. 

0T=  OP  tun  OPT:  that  is. 


Polar  subtangent  =  ?•  tan  \p  = 

0N=^  OP  cot  PNO;  that  is, 
Polar  subnormal  =  r  cot  ip  — 


(W 


154.    Angle  of  Intersection.     Suppose  the  two  curves  intersect  at  P, 
aiul  Lave  the  tangents  PT  and  PI". 
_OPT=xjj,     OPT'  =  xl,'. 

Then  the  angle  of  intersection, 


and       tan  /:: 


tan  i//'  —  tail  t/^ 
1  +  tan  i}/'  tan  i//' 


0) 


By  this  formula  the  angle  of  inter- 
section may  be  found  in  polar  coordi- 
nates, in  the  same  way  as  by  (1),  Art.  147,  in  rectangular  coordinates. 

For  example,  find  the  angle  of  intersection  between  the  curves 


and 


r  =  as\nJU, 
r  =  a  cos  2^. 


(•M 


From  (2)  and  (3)  we  have  for  the  intersection 
tau2d  =  l. 


184 

DIFFERENTIAL   CALCULUS 

From  (2), 

tan </''  =  -  tan 26  =  ~,  for  the  intersection. 

From  (3), 

tan  i//  = cot  2  ^  =  —  -,  for  the  intersection. 

Substituting 

in  (1),                   tan/=-. 

The  curves  are  that  in  Art.  144,  and  the  same  curve  revolved  45° 
about  the  origin. 

EXAMPLES 

.  1.   In  the  circle  (Art.  135),  r  =  asin^,  find  i/^  and  <^. 
/  Ans.    ^  =  ^,  and  <^  =  2  ^. 

y'  2.    In   the   logarithmic   spiral  (Art.  138),  r  =  e"^,  show  that  ij/   is 
constant. 

/    3.    In    the    spiral    of    Archimedes    (Art.  136),    r  =  a9,   show  that 
tan  '/'=<?;  thence  find  the  values  of  i(/,  Avhen  ^  =  2  tt  and  4  tt. 

A71S.  80°  57' and  85°  27'. 
Also  show  that  the  polar  subnormal  is  constant. 

4.  The  equation  of  the  lemniscate  (Art.  143)  referred  to  a  tangent 
at  its  center  is  r'-  =  a- sin  2  6.     Find  if/,  ({>,  and  the  polar  subtangent. 

Ans.    ij/  =  2  6]  (f)=io6;    subtangent  —  a  tan  2^  Vsin  2^. 

5.  In  the  cardioid  (Art.  141),  r  =  a(l  —cosO),  find  </>,  ijj,  and  the 
polar  subtangent. 

Ans.   <^  =  — ;    </'  =  7 ;    subtangent  =  2  a  tan  -  sin^  -• 

6.  Find  the   area  of  the    circumscribed  square  of  the  preceding 
cardioid,  formed  by  tangents  inclined  45°  to  tlie  axis. 


4 


21 


Ans.   ^(2  +  V3)( 


DIRECTION   OF  CURVES.     TAX(;ENTS   AXD   XollMALS     185 


/ 


7.    In  the  folium  of  Descartes  (Art.  127),  r=  -^ "  t^"-" ^  «ec g 

^  l+tau-'d    ' 


show  that  tan  <i  =  t'^"^^-2tang 

2tan'^-l 


8.    Find  the  area  of  the  square  circumscribed  about  tlie  loop  of 
the  folium  of  the  preceding  example. 

Ans.    2</2a-. 


r.. 


9.    Show  that  tlie  spiral  of  Archimedes  (Art.  136),  r  =  ad,  and  the 
reeiprocal  spiral  (Art.  137),  r6  =  a,  intersect  at  right  angles. 


10.  Show  that  the  cardioids  (Art.  141 ), 

r  =  a  (1  -  cos  6),     r  =  b(l  +  sin  6), 
intersect  at  an  angle  of  45°. 

11.  Show  that  the  parabolas  (Art.  139), 

.6  ..6 

r  —  m  sec-  -,     r  =  n  cosec-  - 

2  2' 

intersect  at  right  angles. 

-  12.   Find  the  angle  of  the  intersection  between  the  circle  (Art.  l.">.""»). 
'r  =  a  sin  0,  and  the  curve  (Art.  144),  r  =  a  sin  26. 

Ans.    At  origin  0°;  at  two  other  points,   tan"'  3V3  =  79°()'. 


13.  Find  the  angle  of  intersection  between  the  circle  (Art.  135). 
r  =  2  a  cos  6,   and  the  cissoid  (Art.  125),  r  =  2a  sin  0  tan  6. 

A)ni.    tan"'  2. 

14.  At  what  angle  does  the  straight  line,  r cos 6  =  2a,  intersect  tli<' 
circle  (Art.  135),  r  =  5  a  sin  $  ?  ^^^^^^    tan~'  -. 

15.  Show  that  the  equilateral  hyperbolas  (Art.  142),  r  sin  2  6  =  a\ 
j-^cos  2  6  =  1/,  intersect  at  right  angles. 


186 


DIFFERENTIAL   CALCULUS 


16.  Fiud  the  angle  of  intersection  between  the  circles  (Art.  !./•  , 

r  =  a  sin  ^  +  6  cos  6,     r  =  a  cos  6  -\-b  sin  0. 

Ans.   tan'^^^-^^^ — 
2ab 

17.  Find  the  angle  of  intersection  between  the  lemniscate  (Art.l4.')), 
r-  =  cr  sin  2  6,  and  the  equilateral  hyperbola  (Art.  142),  r-sin  2  9  =  U\ 

Ans.     2  sin-^  -• 
a 

155.    Derivative  of  an  Arc.     Rectangular  Coordinates.    Let  s  denote 
the  length  of  the  arc  of  the  curve  measured  from  any  fixed  point  of  it. 


Then 
We  have 


arc  AP,     A.S  =  arc  PQ. 


sec  QPR : 


PQ 

PR 


Now  suppose  A.K  to  approach  zero,  and  consequently  the  point  Q 
to  approach  P. 
Then 

Lim  secQPi?=secTP7?=sec<^.        ^  x 


PQ^     PQ       arc  Pq 
PR      ^vcPQ       PR    ' 


Lim    ^-^-  =  1, 

arc  PQ       ' 

Lim^Q^Lim^^:^^ 
PR  PR 

T  ■     A.s       (Is 

=  Lim —  = — . 

i\X      (Ix 

fls 


Hence 


'foi'e 


sec  <^  = 


ds 


dx 


'>=V- 


^^=Vl  +  tan^<^=Vl+(| 


Ax 


■>ly 


M  N 


(1) 


DlKliCTIUN    OF   CUUVKS.     TANGENTS    AM)    NORMALS      IS' 


It  is  .evident  also  that 

SUl(j>  =  -^,      cos^= — . 
ds  ds 


C-i) 


It  uuiy  be  noticed  that  these 
relations  (1)  and  (2)  are  cor- 
rectly represented  by  a  right 
triangle,  whose  hypothenuse  is 
ds,  sides  dx  and  dy,  and  angle 
at  the  base  tf). 


Here 


d«=V(fix-)- +  (%)-, 


w^-dj- 


156.   Derivative  of  an  Arc.    Polar  Coordinates.     From  the  figure  of 
Art.  152,  we  have,  as  A^  approaches  zero. 


sec  ^  =  Lini  sec  PQE  —  Lini 


As 

RQ 


As 


J  .      arc  J'Q 
RQ 

As 
Ad 


Lin 


A  /•  +  2  r  sin- 


A^ 


A^ 


Ar 
A^ 


A^ 


A^ 


A.s 

RQ' 


sec  if/  =  Lim 


ds 
A.s  ^(W      ds 
RQ~7[?~dr' 

dO 


Hence  f >  V I+T^Iii^A  =  V ^  +  ^(f )' 


dd     drde      >/         \ddj 


(1) 
(2) 


188 


DIFFERENTIAL   CALCULUS 


It  may  be  noticed  that  these  rehations  (1),  (2),  and  (3),  are  cor- 
rectly represented  by  a  right  triangle,  whose  hypothenuse  is  els,  sides 
dr  and  rdO,  and  angle  between  dr  and  ds,  if/. 


Here  ds  =  V  {drf  -j-  (/•  c/^)-, 

and  thence 


ds         L    ,     .JdOy  ds  „  ,   /drV 


CHAPTER   XVI 
DIRECTION  OF  CURVATURE.     POINTS   OF  INFLEXION 

157.  Concave  Upwards  or  Downwards.  A  curve  is  .said  to  be  con- 
cave iqjicards  at  a  point  P,  when  in  the  iinniediate  neighborhood  of 
P  it  lies  wholly  above  the  tangent  at  P,  as  in  the  first  figure  below. 
Similarly,  it  is  said  to  be  concare  doicnicards,  when  in  the  immediate 
neighborhood  of  P  it  lies  wholly  below  the  tangent  at  P,  as  in  the 
second  figure  below. 

It  will  now  be  shown  that  when  the  equation  of  the  curve  is  in 
rectangular  coordinates,  the  curve  is  concave  V2>n-anls  or  doioncards, 
according  as  — -,  is  positive  or  )U'(iatir" 


X    0 


Suppose  ^'  >  0,  that  is,  jLf^^\  >  0:  in  other  words,  the  derivative 
dx-  dx\dxj 

of  the  slope  is  positive. 

Then  by  Art.  21  the  slope  increases  as  x  increases. 

This  case  is  illustrated  in  the  first  figure  above,  where  the  slope 
evidently  increases  as  we  pass  from  /*,  to  P..  The  curve  is  then  con- 
cave upwards. 

But  if  ^<  0,  it  follows  that  the  slope  decreases  as  x  increases. 
daf 

18U 


190 


DIFFERENTIAL   CALCULUS 


We  then  have  the  case  of  the  second  figure,  where  the  slope  cle- 
creases  as  we  pass  from  P^  to  Pg-  The  curve  is  then  concave  doivn- 
ic((rcls. 


158.     A  Point  of  Inflexion  is  a  point  P  where  -^  changes    sign, 

dx- 

tha  curve  being  concave  upwards  on  one  side  of  this  point,  and  con- 

LMve  downwards  on  the  other. 

This  can  occur,  provided  —  and 
dx 

^  are  continuous,  only  when 
d'jt? 

^  =  0.     . 
dx^ 


(1) 


Bat  if  ^  and  ^'^  ^^- 


are  infinite,  we 


dx         dx- 
may  have  a  point  of  inflexion 


when 


dx- 


It  is  evident  that  the  tangent  at 'a  po^t  of  inflexion  crosses  the 
curve  at  that  point. 

For  example,  find  the  point  of  inflexion  of  the  curve 
22/  =  2-8x  +  6x2-iB3. 


Here 


^  =  3(2-.t). 
dx- 


Putting  this  equal  to  zero,  we  have  for  the  required  point  of  in- 
flexion, cc  =  2.    If  .r  <  2,  ^^  >  0 ;  and  if  x  >  2,  ^,  <  0. 
dx-  dxr 

Hence  tlie  curve  is  concave  upwards  on  the  left,  and  concave  down- 
wards on  the  right,  of  the  point  of  inflexion. 


DIRECTIOX   OF   CURVATURi:.      POINTS   OF    IXFI.KXIOX      191 


\,j:  EXAMPLES 

Find  the  points  of  inflexion  and  the  direction  of  curvature  of  the 
five  following  curves: 

V       1.     y=(x'-lf. 

Ans.     x=± — r^;  concave  downwards  between  these  points,  con- 
V3 
cave  upwards  elsewhere. 

O      2.     y  =  x'  -  16  x"  +  42  x-  -  28  x. 

Ans.     a'=l  and  x=7;  concave  downwards  between  these  points, 
concave  upwards  elsewhere. 


3.     cc^ij  =  X  (x  —  ay  +  a\v. 

Ans.     x=^^^^;    concave  downwards  on  the  left  of  this  point,  con- 
o 

cave  upwards  on  the  right. 


4.    The  witch  (Art.  126),  **=  ^-^-r.' 
.1--  +  4  a- 

Ans    (  ± -—  —  1 ;    concave    downwards    between  these  points, 

V    V3   -V 

concave  upwards  outside  of  them. 


X 

5.    The  curve,  ?/=  — — -— -• 

Ans.     f-3a,  -— \   (0,0),   fsn,   ^\,  concave  upwards  on  the 

left  of  iirst  ])()int,  downwanls  l.etwcen  first  aiul  second, 
upwards  between  second  and  third,  and  downwards  on  the 
right  of  third  point. 


192  DIFFERENTIAL   CALCULUS 

Find  the  points  of  iftflexion  of  the  following  curves: 

6.   y  =  -^--  Ans.    a;  =  0and±2V3. 

ar +  4 

7  g  a;  ^^g^     x  =  —2a. 

I 

'       ^  "    ■    133)  _      _ 

^  V2 


^ns.     cc  =  0  and  .t  =  3. 


j_„,,     .x  =  2aog«-log6). 


/in^)ny=^^"-' 


11.   aV  =  aV  -  a;«  (Art.  134).  Ans.     a;  =  ±  ^  ^27  -  3  V33. 


\ 


CHAPTER    XVir 

CURVATURE.  RADIUS  OF  CURVATURE.  EVOLUTE  AND 
INVOLUTE 

159.  Curvature.  If  a  point  moves  in  a  straight  line,  the  direc- 
tion of  its  motion  is  the  same  at  every  point  of  its  course,  but  if  its 
path  is  a  curved  line,  there  is  a  continual  change  of  direction  as  it 
moves  along  the  curve.     This  change  of  direction  is  called  curvature. 

We  have  seen  in  the  preceding  chapter  that  the  sign  of  the  second 
derivative  shows  which  way  the  curve  bends.  We  shall  now  find 
that  the  first  and  second  derivatives  give  an  exact  measure  of  the 
curvature. 

.  The  direction  at  any  point  being  the  same  as  that  of  the  tangent 
at  that  point,  the  curvature  may  be  measured  by  comparing  the 
linear  motion  of  the  point  with  the  simultaneous  angtdar  motion  of 
the  tangent. 

160.  Uniform  Curvature.  The  curvature  is  uniform  when,  as  the 
point  moves  over  equal  arcs,  the  tangent  turns  through  equal  angles. 
The  only  curve  of  uniform  curvature  is  the  circle.  Here  the  meas- 
ure of  curvature  is  the  ratio  between  the  angle  described  by  the  tan- 
gent and  the  arc  described  by  the  point  of  contact.  In  other  words, 
it  is  the  angle  described  by  the  tangent  while  the  point  describen  a  unit 
of  arc. 

Suppose  the  point  Pto  move  in  the  circle  AQ. 

Let  s  denote  its  distance  AP  from  some  initial  position  .1,  ami 
<^  the  angle  PTX  made  by  the  tangent  PT  with  OX. 

Then  as  the  point  moves  from  P  to  Q,  s  is  increased  by  PQ  =  Ax, 
and  <f)  by  the  angle  QRK=  A</>. 

As  the  point  describes  the  arc  As,  the  tangent  turns  tlirough  the 
angle  A<^. 


194 


DIFFERENTIAL   CALCULUS 


The  curvature,   being  uni- 


form, is  then  equal  to 


Acf> 
As' 


If  we  draw  the  radii  CP, 
CQ,  and  let  r  denote  the 
radius,  theu 

angle  PCQ  =  QBK=  A</>. 

But 
arc  PQ  =  CP  (angle  PCQ) ; 


that  is, 


rA<^, 


A<^_1 


Hence  the  curvature  of  a  circle  is  the  reciprocal  of  its  radius. 
For  example,  suppose  the  radius  of  a  circle  to  be  50  feet. 


Then  its  curvature  is 


A<^^  1 
As  ~50' 


where  A<^  is  in  circular  measure,  and  As  in  feet. 

In  other  words,  for  every  foot  of  arc,  the  change  of  direction  is 


50 


in  circular  measure  =  1°  8'  45". 


161.  Variable  Curvature.      For  all  curves  except  the   circle  the 
curvature  varies  as  we  move  along  the  curve.     In  moving  over  the 

arc  A  5,  — ^  is  the  mean  curvature  throughout  the  arc.     The  curva- 
As 

ture  at  the  beginning  of  this  arc  is  more  nearly  equal  to  — ^,  the 

shorter  we  take  A.s. 

Hence  the  curvature  at  any  point  of  a  curve  is  equal  to 

As       ds 


CUKVATURE.      RADHS   OF   CURVATrilK 


195 


162.     Circle  of  Curvature.     A  circle  tangent  to  a  curve  at  any  point, 

having  its  concavity  turned  in  the  same  direction,  and  having  the 
same  curvature  as  that  of  the  curve  at  that  point,  is  called  the  circle 
of  curvature;  its  radius,  the  radius  of  curvature ;  and  its  centre,  the 
centre  of  curvature. 

The  figure  shows  the  circle  of  curvature  MPX  for  the  point  P  of  the 
ellipse.    C  is  the  centre  of  curvature,  and  CP  the  radius  of  curvature. 

It  is  to  be  noticed 
that  the  circle  of  curv- 
ature crosses  the  curve 
at  P.  This  can  be 
easily  proved. 

At  P  the  circle  and 
ellipse  have  the  same 
curvature,  but  as  we 
go  towards  P,,  the 
curvature  of  the  ellipse 
increases,  while  that  of 
the  circle  continues  the  same. 

Hence  on  the  right  of  P  the  circle  is  outside  of  the  ellipse. 

Moving  from  P  to  Po,  the  curvature  of  the  ellipse  decreases,  and 
therefore  on  the  left  of  P  the  circle  is  inside  of  the  ellipse. 

So  in  general  the  circle  of  curvature  crosses  the  curve  at  the  jioint  of 
contact. 


B 

>?. 

~~7~^ 

(       / 

/     p1^ 

N, 

v/ 

0 

JSl^ 

> 

L 

196  DIFFERENTIAL   CALCULUS 

The  only  exceptions  to  this  rule  are  at  points  of  maximum  and 
minimum  curvature,  as  the  vertices  A  and  B  of  the  ellipse. 

As  we  move  from  A  along  the  curve  in  either  direction,  the  curva- 
ture of  the  ellipse  decreases ;  hence  the  circle  of  curvature  at  A  lies 
entirely  within  the  ellipse. 

Similarly  it  appears  that  the  circle  of  curvature  at  B  lies  entirely 
without  the  ellipse. 


163.     Radius  of  Curvature.      The  curvature  of  the  circle  of  curva- 
ture being  that  of  the  given  curve,  is  equal  to  -2    (Art.  161).    If  we 

denote  the  radius  of  curvature  by  p,  then  by  Art.  160, 

ds  ... 

P  =  di ^^^ 

To  obtain  p  in  terms  of  x  and  y,  we  may  write  (1),  ^  =  ^  =  -1.  • 

dfj)      d(f) 


dx 


From  (1)  Art.  155, 


=ViH2)- 


Also,  tan<)!>  =  ^^    </>  =  tan-Y^ 

dx  \dx 


d?y 

dx' 

1+7^ 


.     Differentiating^  ^  = ^^^, (2) 

^  dx      ^    ,     dy\ 


\!^m 


Hence  P= ^/^  (^) 

dx' 


CURVATURE.  RADIUS  OF  CURVATURE       197 

It  is  to  be  noticed  that  p  is  always  to  be  considered  positive ;  that 
is,  the  sign  of     1  4-  (^~''\\    is  taken  the  same  as  that  of  '-^^ . 
By  interchanging  x  and  ?/,  we  have 


\}^mf 


cly- 

which  is  sometimes  the  more  convenient  expression. 

As  an  example,  find  the  radius  of  curvature  of  the  semicubical 
parabola  ay-  =  x^  (Art.  130). 

Differentiating         ^^-^"^      <^-       ^ 


dx     „    ^'    dx-      .  /     ^.  A^ 


Substituting  in  (3),  we  find 


x2(4a  +  9  a.')  2 
P= 6^ 


164.   Radius   of   Curvature  in   Polar   Coordinates.      Kesuming  (1), 

Art.  163,  p  =  ~,   let  us  express  p  in  terms  of  r  and  6. 
d<i> 

,„  .^  r/.s      de 

We  may  wnte  o  =  —  =  -:-  • 

de 

ds         I  o  ,  fdrV 


From  (3),  Art.  156,        %=,j''  +  ('^,] 


From  (2),  Art.  152, 


*=^+*'  ••■^^>+i^• 


198  DIFFERENTIAL   CALCULUS 

From  (1),  Art.  152, 

'/•  ,  f  r  1 

(If  dr 

dO  [dd\ 


Differentiating, 

#    \<iej      <ie' 

Substituting, 

c?</,         "  \dd)      de' 

Hence 

(1) 


EXAMPLES 
Find  the  radius  of  curvature  of  the  following  curves  : 

1.  y==(^x-iy(x-2),   at  (1,  0)  and  (2,  0).  Ans.  p  =  ~  and  --^. 

2.  y=  log  X,  when  x=^.  Ayis.  p  =  2f  | 

(a*  4- 9  x'^Y 

3.  The  cubical  parabola  (Art.  130),  a-y  =  x\      Ans.  p  =  ^ — — j 

s 

4.  The  parabola,  y"  =  4aa;.  ^«s.  p  =  -^^^ — -^ 

Find  the  point  of  the  parabola  where  p  =  54  a.  Ans.  x  =  Scl 

(x-  +  v^Y 
5    The  equilateral  hyperbola,     2xy  =  a\  Ans.  p  =  ^^^ ^^— 


CURVATURE.   RADIUS  OV   CURVA'I  I  KK       IK!) 

6.  The  ellipse,  "^  +  ^  =  1.  Aus  p=l«V  +  6V;}l. 

What  are  the  values  of  p  at  the  extremities  of  the  axes? 

^-l/(«.  -  and 

a  h 

7.  Show  that  the  radius  of  eurvature  of  the  curve, 

^  +  ir+  10  a;  —  4  ?/  -f-  20  =  0  is  constant,  and  equal  to  \\. 

Find  the  radius  of  curvature  of  the  following  curves: 

8.  V  +  log  (1  -  X-)  =  0.  Ans.  p  =  -i^-±^'. 

2  (1  -  j^) 

9.  siny  =  e\  Ans.  p  =  e"'. 

10.  The  catenary  (Art.  128),  v  =  -(J  +  e  '«).  Ans.  p  =  ^  . 

2  a 

11.  The  hypocycloid  (Art.  132),  x^  + 1/  ==  u\         Ans.  p  =  3(axy)^- 

12.  The  curve  aV  =  «'-^*-^-'' (Art.  13.3),  at  the  points  (0,  0)  and 
(a,  0).  Ans.  p=  -  and  p  =  a. 

13.  The  cycloid,  a;  =  a($  —  sin $),  y  =  a{\—  ccs^j. 

♦  ^l/*.s.  p  =  4a  sin  - 

14.  Show  that  the  radius  of  curvature  of  the  logarithmic  spiral 
(Art.  138),  r  ■=  e"*,  is  proportional  to  r.  p  =  '"  Vl  +  «'• 

15.  Show  that  the  radius  of  curvature  of  the  curve  (Art.  135), 

1 


r  =  a  sin  6  -{-  h  cos  6,  is  constant.  P  =  o  ^"  "'" 

^         16.    The  spiral  of  Archimedes  (Art.  136),  r  =  a^.  , 


200 


DIFFEKENTIAL   CALCULUS 


17.    The  cardioid  (Art.  141),  r  =  a  (1  -  cos  0). 


18.  The  curve,  r  =  a  sin''  f  (  Art.  145). 

o 

19.  The  imrabola  (Art.  139),  r  =  a  SQor  f- 

20.  The  lemniscate  (Art.  143),  r^^a^  cos  2^. 


Ans.  p"  =  -  cir. 

.  3      .   .6 

Ans.  p=-a  sin- 
4  3 

a 

Ans.  p  =  2a  sec''  -  ■ 


Ans.  p  = 


3r 


165.  Coordinates  of  the  Centre  of  Curvature.  Let  x,  y  be  the  co- 
ordinates of  P,  any  point  of  the  curve  AB,  and  C  the  corresponding 
centre  of  curvature.  CP  is 
then  the  radius  of  curvature, 
and  is  normal  to  the  curve. 

Draw  also  the  tangent  PT. 

Then       CP=  p; 
angle  PCM  =  PTX  =  cf>. 

Let  a,  j3,  be  the  coordinates 
of  a     OL  =  OM-  RP, 
LC^MP+RC; 

that  is,    a  =x  ~  p  sin  ^, 

^  =  2/  +  /3eosc^.       (1) 

To  express  a  and  fi  in  terms  of  x  and  y,  we  have,  by  (2),  Art.  15;"), 
and  (1),  (2),  Art.  163, 


p  sin . 


p  cos  <^ : 


(ls^djl_  dy  _  dy  dx  _  dx  _ 
d(f>  ds      d<f>     dx  d<f> 


1  + 


dyV 
dx 


ds  dx _dx 
dff)  ds      d(f> 


-(f 


dry 
■  dx" 


d'-y 
dx- 


Hence  dy 

dx 


Hm 


cPy 
dx^ 


y  +  - 


d^ 
dx- 


(2) 


CURVATURE.  RADIUS  OF  CURVATURE 


201 


166.    Evolute  and  Involute.      Every  point  of  a  curve  AB  has  a 


Thus,  P„  P.,  Pj,  etc.,  have  for 


corresponding  centre  of  curvature, 
their  respective  centres  of 
curvature  Cj,  Cj,  C3,  etc.  The 
curve  HK,  which  is  the  locus 
of  the  centres  of  curvature,  is 
called  the  evolute  of  AB.  To 
express  the  inverse  relation, 
AB  is  called  the  involute 
of  HK 


167.  To  find  the  Equation 
of  the  Evolute  of  a  Given  Curve. 
By  (2),  Art.  165,  a  and  /8,  the 

coordinates  of  any  point  of  the  required  evolute,  may  be  expressed 
in  terms  of  x  and  y,  the  coordinates  of  any  point  of  the  given  curve. 
.These  two  equations,  together  with  that  of  the  given  curve,  furnish 

three  equations   between  «,  /3,  x, 


and  y,  from  which,  if  x  and  y  are 
eliminated,  we  obtain  a  relation 
between  a  and  /?,  which  is  the 
equation  of  the  required  evolute. 

For  example,  find  the  equation 
of   the   evolute    of    the   parabola 

?/-  =  4  ax. 


Here 

dy_ 
dx 
iPy 
dx- 

Substituting 

in  (2),  Art. 

165, 

w 

have 

a  — 

202 


DIFFERENTIAL   CALCULUS 


Eliminating  x,  we  have  for  the  equation  of  the  evolute, 

This   curve  is   the  semicubical   parabola  (Art.  130).      The  figure 
shows  its  form  and  position.     F  is  the  focus  of  the  given  parabola. 
OC=2a  =  2  0F. 

As  another  example,  let  us  find  the  equation  of  the  evolute  of  the 
ellipse,  ,2 


a-  0' 

dy  _  _  6^^  r?-?/ 

dx          d-y  dx' 
Substituting  in  (2),  Art.  165, 

a* 


ay 


(Art.  66) 


b* 
To  eliminate  x  and  y  be- 
tween these  equations  and 
that  of  the  ellipse,  we  find 

^  _      ha        y^  bff 


-b-     b'         a?-b- 

^2     r  ^  {aaf+{bj3,)i  ^^ 
a-      b'         (a^-lj^f 

giving,  for  the  equation  of 
the  evolute, 

{auf  +  {bpy=  {d'-b-f. 

The  evolute  is  EF'E'FE. 
E  is  centre  of  curvature 
for  ^;  CforP;  i^^fori?; 
E'  for  ^';  F'  for  B'. 

In  the  figure  F  and  F' 
are  outside  the  ellipse, 
but     if    the    eccentricity 


CURVATURE.  RADIUS  OF  CURVATUKi: 


m 


is    decreased,   so    that    a<b^'J,   these    points    full    within    tlie 
ellipse. 

168.    Properties  of  the  Involute  and  Evolute.     Let  us  return  to  tlie 
equations,  (1),  Art.  165, 

a  =  X  —p  sin  <^, 
(3  =  )/ +  p  cos  (f). 

Differentiating  with  respect  to  s, 

da     dx     dp    .     ,  ,(Jdi 

f  =  'i'  +  ^Poos*-psin*l^.      ...     (2) 
ds      ds      ds  ds  ^  ^ 

•     Substituting  in  (1),      p  =  ~  and  cos  <^  =  — ,  (Art.  155),  two  terms 

d(j)  ds 

cancel  each  other,  giving 


da  dp   ■      .  ,„. 


Similarly  in  (2),  /o  =  —  and  sin  <^=  '-^  (Art.  155),  giving 

(/<^  ds 

^  =  ^cos<^ (4) 

ds      ds       ^  ^  ^ 

Dividing  (4)  by  (3),  ^^  =  __1^ (5) 

dti  tan  (ft 

But  -i-  is  the  slope  of  tlie  tangent  to  the  evolute  at  any  point  C',, 
du 
(see  fig..  Art.  IGG),  and  tan  <^  the  slope  of  the  tangent  to  the  involute 
at  the  corresponding  point  Pj.  Since  by  (5)  one  is  minus  the  recijv 
rocal  of  the  other,  these  tangents  are  perjjendicular  to  eacli  other. 
In  other  words,  a  tangent  to  the  evolute  at  any  point  C\  is  C\J\,  the 
normal  to  the  involute  at  J\. 


204  DIFFERENTIAL   CALCULUS 


169.    Again,  from  (3)  and  (4),  Art.  168, 


^(fJHIJ"'-(l7K.^> 


where  s'  denotes  the  lengtli  of  the  arc  of  the  evolute  measured  from 
a  fixed  point.     Hence, 

Hence,  s'  ±  p  =  a  constant, (1) 

since,  if  a  derivative  is  always  zero,  the  function  can  neither  increase 
nor  decrease,  but  is  constant. 
It  follows  from  (1)  that 

A(s'±p)  =  0,     As'  =  ±^p. 

That  is,  the  difference  between  any  two  radii  of  curvature  PjCj, 
P3C3,  is  equal  to  the  corresponding  included  arc  of  the  evolute  C1C3. 

170.  From  the  two  properties  of  Arts.  168  and  169,  it  follows  that 
the  involute  AB  may  be  described  by  the  end  of  a  string  luiwound 
from  the  evolute  HK.  From  this  property  the  word  evolute  is 
derived. 

It  will  be  noticed  that  a  curve  has  only  one  evolute,  but  an  infinite 
number  of  involutes,  as  may  be  seen  by  varying  the  length  of  the 
string  which  is  unwound. 

EXAMPLES 

1.    Find  the  coordinates  of  the  centre  of  curvature  of  the  cubical 

parabola  (Art.  130),  a'y  =  x\  4  + 15  ^.4  ^,^  _  9  ,^.5 

Ann.    a  =  — ^!^— ,     B  =  — - — -. — . 


2.    Find  the  coordinates  of  centre  of  curvature  of  the  semicubical 
parabola  (Art.  130),  ay-  =  x^.  _ 

An.    «  =  — If,     ^  =  ^(^-  +  ^)Vl- 


CURVATURE.      RADIUS   OF   CUKVATriiK  205 

3.  Find  the  coordinates  of  the  centre  of  eurviitiue  of  the  catenary 

(Art.  128),2/  =  ?(j4-e-»).  

Ans.    «  =  .r-^v>'-a^    ^  =  2,,. 

4.  Show  tliiit  in  the  parabola  (Art.  129),  x^  +  y-  =  a-,  we  liave  tlie 
relation  a-\-  (S  —  o  (x  +  //). 

5.  Find  the  coordinates  of  the  centre  of  curvature,  and  the  equa- 
tion of  the  evolute,  of  the  hypocycloid  (Art.  132),  x^^  +  >/^  =  aK 

Ans.    a=  a  +  3  x^  y^,     /3  =  »/  +  3  x^  y', 

6.  Given  the  equation  of  the  equilateral  hyperbola  2xy  =  a^, 

show  that  «  +  /?  =  ^^'■^■''^\     a -(3=  ^Jl^=^. 

a-  d- 

Thence  derive  the  equation  of  the  evolute, 

(«+y8;^-(a-/3)^=2a3. 

7.  Find  the  equation  of  the  evolute  of  the  cissoid  (Art.  12n), 

«2=_if_.  Ans.    409GaV-f  1152  (ry8=^+ 27)3*  =  0. 

2  a  —  ic 


CHAPTER.  XVIII 

ORDER  OF   CONTACT.     OSCULATING  CIRCLE 

171.    Order  of  Contact.     Let  us  consider  two  curves  whose  equa- 
tions are 

y  =  (l>(x)     and     y  =  i/'(x). 

If  for  a  definite  value  a,  of  x,  the  value  of  y  is  the  same  for  both 
curves,  that  is,  if 

<j>(a)  =  xj,(a), 

the    curves     have     a     common 
point  F. 

If,   moreover,   for   x  =  a,    the 

value  of  —  also  is  the  same  for 
dx 

both  curves,  that  is,  if 

^(a)  :=:  i//(a)  and  <j)'(a)  =  if/'(a), 

the  curves  have  a  common  tangent  at  P. 

The  curves  are  then  said  to  have  a  contact  of  the  Jirst  order. 

If  besides,  for  x  =  a,  the  values  of  ^    are    the    same    for  both 
curves,  that  is,  if 

<t>(a)=^(a),     <^'(a)  =  ^'(«).    ^^d    </>"(a)  =  <(a), 

the  curves  have  contact  of  the  second  order. 

In  general,  the  conditions  for  a  contact  of  the  ?ith  order  at  the 
point  X  ■=  a,  are 

<f>(a)  =  ip(a),      (/>'((/)  =V'' (a),      </>"(«)  =  ^"  (a),     •-,     .^"(a)  =  f  (a), 
and  <^"+^  (rt)  T^  ,^"+'  (a). 

206 


{ 


ORDER   or   CONTACT.     OSCTLATIXO   CIRCLE  207 

In  other  words,  for  x  =  a, 


dy     d^ 
dx     dx'' 


dx-'' 


must  all  have  the  same  values,  respectively,  taken  from  the  equations 

of  both  curves ;  and ■'  must  have  different  values. 

dx"+^ 


172.  When  the  Order  of  Contact  is  Even,  the  Curves  cross  at  the  Point 
of  Contact ;  but  when  the  Order  is  Odd.  they  do  not  cross.  Let  us  dis- 
tinguish the  ordinates  of  the  two  curves  by 

Y=<f>(x),     and    y  =  }p(x). 

In  the  figures  Prefers  to  the  full  curve,  and  y  to  the  dotted  curve 
If   Y—y  has  the  same  sign  on  both  sides  of  P,  as  in  the  first 
figure,  the  curves  do  not  cross  at  P;  but  if  Y—y  is  positive  on  one 
side  of  P  and  ne'^ative  on  the  other,  the  curves  do  cross  at  P. 


Let 
Then 


P.Q. 


0M=  a,     MM,  =  h. 

Y -y  =  ^{a  +li)  -  4,{a  +  h). 


1 

p 

^— — 

' 

yi^ 

Q. 

F 

) y^ 

Qy/ 

/ 

/ 

1 

-7 

p= 

0             \ 

v^,             \ 

^        ^ 

^i 

~~x 

Expanding  by  Taylor's  Theorem, 


]r 


P,Q,  =  <i>(a)  +  h<l>'(a)  +  '-<^"('0  +  -  0"'(«)  +  •• 


^(a)  -  h.p '  (a)  -  |V"(a)  -  ||V"'(«)  " 


208  DIFFERENTIAL   CALCULUS 

Suppose  the  contact  of  the  first  order ;  then 

cf) (a)  =  ij/ (a),     <}>'((()  =  ip'(ci),     and  (1)  becomes 


Pi 


Qi=,fr</>''(«)-«A'x«)]+|r'^''x«)-'A'''(«)]+----  •  (2) 


For  sufficiently  small  values  of  h  the  sign  of  the  lowest  power  de- 
termines that  of  the  second  member,  and  hence  the  sign  of  PiQi  will 
remain  unchanged  when  —h  is  substituted  for  h,  giving  PoQ-,,  as  in 
the  first  figure. 

Thus  when  the  contact  is  of  the  first  order,  the  curves  do  not  cross 
at  the  point  of  contact. 

Again,  suppose  the  contact  of  the  second  order  ;  then 

<ji"{a)  —  ^li"{a),  and  (2)  becomes 
PiQi  =  || r<^  '"(«)  -  ^"'(«)]  +1^  \^\a)  -  r{a)^  +  •••• 

Now  PiQi  will  change  sign  with  h,  so  that  P2Q2  and  PiQi  will  have 
different  signs,  as  in  the  second  figure. 

Thus  when  the  contact  is  of  the  second  order,  the  curves  cross  at 
the  point  of  contact. 

By  similar  reasoning  the  general  proposition  is  established. 

It  may  be  of  service  to  the  student,  in  connection  with  this  prin- 
ciple, to  think  of  two  curves  as  having  two  consecutive  common 
points,  when  they  have  contact  of  the  first  order;  as  having  three 
consecutive  common  points,  when  they  have  contact  of  the  second 
order;  as  having  ?t  +  l  consecutive  common  points,  when  they  have 
contact  of  the  nth  order. 

An  odd  number  of  common  points  implies  the  crossing  of  the 
curves,  but  where  tliere  is  an  even  number  of  common  points,  the 
curves  do  not  cross. 

173.  Osculating  Curves.  Contact  of  the  nth  order  requires  that  y 
and  its  first  n  derivatives  should,  for  some  definite  value  of  x,  have 
the  same  values  for  both  curves. 

This  implies  n .  + 1  conditions. 


ORDER   OF    CONTACT.      OSCULAllM;    lIKCLl':  20'J 

The  equation  of  the  straight  line,  y  =  (u-  +  h,  having  only  two 
arbitrary  constants,  can  satisfy  only  two  of  these  coiulitions.  Hence 
a  straight  line  can  have  contact  of  the  first  order  with  a  given  curve, 
and  cannot,  in  general,  have  contact  of  a  higher  order. 

The  equation  of  the  circle  x-  +  y-  +  ax  +  by  +  c  =  0,  having  three 
arbitrary  constants,  can  satisfy  three  of  the  conditions.  Hence  the 
circle  may  have  contact  of  the  second  order  with  a  given  curve. 
Such  a  circle  is  called  the  osculatuig  circle. 

Similarly,  the  parabola,  whose  equation  contains  four  constants, 
may  have  contact  of  the  third  order ;  and  the  general  conic,  whose 
equation  contains  five  constants,  may  have  contact  of  the  fourth 
order  with  a  given  curve.  These  are  called  the  osculating  parabola 
and  the  osculating  conic. 

174.  Order  of  Contact  at  Exceptional  Points.  Although  the  tangent 
has  generally  contact  of  the  first  order,  it  may  at  exceptional  points 
of  a  curve  have  a  contact  of  a  higher  order. 

For  example,  since  the  tangent  at  a  point  of  inflexion  crosses  the 
curve,  it  follows  from  Art  172,  that  the  order  of  contact  must  be 
even.  Hence  at  a  point  of  inflexion  the  tangent  has  contact  of  at 
least  the  second  order. 

The  osculating  circle,  which  has  generally  contact  of  the  second 
order,  has  a  higher  order  of  contact  at  points  of  maximum  or  mini- 
mum curvature,  as,  for  example,  the  vertices  of  an  ellipse.  It  is 
evident  from  the  symmetry  of  the  ellipse  with  reference  to  its  ver- 
tices, that  no  circle  tangent  at  these  points  would  cross  the  curve  at 
the  point  of  contact.  Hence,  by  Art.  172,  the  order  of  contact  is 
odd,  —  at  least  the  third. 

175.  To  Find  the  Coordinates  of  the  Centre,  and  Radius,  of  the  Oscu- 
lating Circle  at  Any  Point  of  a  Given  Curve. 

Let  the  equation  of  the  given  curve  be 

The  general  equation  of  a  circle  with  centre  (a,  b)  and  radius  r,  is 
(x-ay+(y-by-=r (1) 


210  DIFFERENTIAL   CALCULUS 

Differentiating  twice  successively,  we  have 

dx 


1  + 


(^^^-^%-'' 


(2) 
(3) 


From  (3), 


h  = 


^-m 


dx' 


(4) 


From  (2), 


[-(2)] 


d-y 
d5 


(5) 


Substituting  (4)  and  (5)  in  (1), 


7^  = 


(6) 


Hence 


1  + 


1  + 


da? 


,     b==y  + 


^dy^ 
Ax  J 


d?y 
,    da? 


,      (T) 


and 


1  + 


Wl 


da? 


(8) 


ORDER   OF    CONTACT.      OSCl LATINC.    CIRCLK  -^11 

In  these  expressions,  x,  y,-^,  --^,  refer  to  (1),  the  eriuatiun  of  tlie 
dx   dx' 

circle  ;  but  since  the  osculating  circle  by  definition  lias  contact  of  the 

second  order  with  the  given  curve,  these  quantities  will  have  the 

same  values  if  derived  from  the  equation  of  this  curve  y=f(^x),  and 

applied  to  the  point  of  contact. 

By  comparing  (7)  and  (8)  with  the  expressions  for  «,  fi,  and  p,  in 

Arts.  103,  165,  it  is  evident  that  the  osculating  circle  is  the  same  as 

the  circle  of  curvature. 

176.  At  a  Point  of  Maximum  or  Minimum  Curvature,  the  Osculating 
Circle  has  Contact  of  the  Third  Order. 

If  we  regard  equation  (8)  in  the  preceding  article  as  referring  to 
the  given  curve  ?/  =f(x),  we  have  as  a  condition  for  a  maximum  or 
minimum  value  of  r, 


dx 
"We  thus  obtain  from  (8), 


'^^0. 


dx\dxV      L  ^\dxj]d.r'       ' 


P^dy/cfyV 
from  which  —4  =  — —ri — r (^) 

Again,  if  we  regard  (8)  as  referring  to  the  osculating  circle 
(x-a)'+(y-by=r% 

we  shall  also  have  -r  =  ^> 

ax 

since  r  is  constant  for  all  points  on  the  circle. 


212  DIFFERENTIAL   CALCULUS 

Thus  we  obtain,  both  for  the  curve  and  the  circle,  the  same  ex- 
pression (1)  for  -^ ,  and  since  -^  and  — ^  in  the  second  member  of 
^  ^  ^        dor  dx  dx" 

(1)  have,  at  the  point  of  contact,  the  same  values  for  both  curves,  it 

follows  that  —^  has  likewise  the  same  value.     Hence  the  contact  is 
dx- 

of  the  third  order. 

EXAMPLES 

1.    Find  the  order  of  contact  of  the  two  curves, 

y  =  x",     and     ?/  =  3  .x-  —  3  a-  +  1. 

By  combining  the  two  equations,  the  point  x  —  1,  y  =  1,  is  found 
to  be  common  to  both  curves. 

Differentiating  the  two  given  equations. 


When 


y  =  x^, 

2/  =  3.t2-3.x  +  1, 

^  =  3a;^ 
dx 

dx 

S=- 

dx^       ' 

S-' 

d^ 

x=l, 

dy_ 
dx 

--  3,  in  both  curves ; 

x  =  l, 

dh, 
dx" 

=  G,  in  both  curves ; 

when 


d^v 
but   —4  has  different  values  in  the  two  curves. 
dx^ 

Hence  the  contact  is  of  the  second  order. 

2.    Find  the  order  of  contact  of  the  parabola,  4:y=x^,  and   the 
straight  line,  y  =  x  —  l.  Ans.    First  order. 


ORDER   OF   CONTACT.     OSCULATING   CIRCLE  213 

3.  Find  the  order  of  contact  of 

9y  =  x^-3x--\-27,  and    9y  +  3x  =  28. 

Ans.    Second  order. 

4.  Find  the  order  of  contact  of  the  curves 

y  =  \og(x  —  l),  and  x^ -Qx +  2y  -\- S  =  0, 
at  the  common  point  (2,  0).  A7is.    Second  order. 

5.  Find  the  order  of  contact  of  the  parabola,  4y  =  ar  —4,  and  the 
circle,  x^  +  y-  —  2y  =  3.  Ans.    Third  order. 

6.  What  must  be  the  value  of  a,  in  order  tliat  the  parabola, 

y  =  x  +  l  +  a(x-iy, 

may  have  contact  of  the  second  order  with  the  hyperbola, 

xy  =  3x  —  l?  Ans.    a= —1. 

7.  Find  the  order  of  contact  of  the  parabola, 

(x-2ay+iy-2ay=2xy, 
and  the  hyperbola,    xy  =  cr.  Ans.    Third  order. 


CHAPTER    XIX 
ENVELOPES 

177.  Series  of  Curves.  When,  in  the  equation  of  a  curve,  different 
values  are  assigned  to  one  of  its  constants,  the  resulting  equations 
represent  a  series  of  curves,  differing  in  position,  but  all  of  the  same 
kind  or  family. 

For  example,  if  we  give  different  values  to  a  in  the  equation  of 
the  parabola  2/^  =  4  ax,  we  obtain  a  series  of  parabolas,  all  having  a 
common  vertex  and  axis,  but  different  focal  distances. 

Again,  take  the  equation  of  the  circle  (x  —  a)^  +  (}/  —  ^)"  =  c-.  By 
giving  different  values  to  a,  we  have  a  series  of  equal  circles  whose 
centres  are  on  the  line  y  =  b. 

The  quantity  a  which  remains  constant  for  any  one  curve  of  the 
series,  but  varies  as  we  pass  from  one  curve  to  another,  is  called  the 
parameter  of  the  series. 

Sometimes  two  parameters  are  supposed  to  vary  simultaneously, 
so  as  to  satisfy  a  given  relation  between  them. 

Thus,  in  the  equation  of  the  circle  {x  —  af  -\-  (y—  h)'  =  c^,  we  may 
suppose  a  and  h  to  vary,  subject  to  the  condition, 

a-  +  &2  =  k\ 

"We  then  have  a  series  of  equal  circles,  whose  centres  are  on 
another  circle  described  about  the  origin  with  radius  k. 

178.  Definition  of  Envelope.  The  intersection  of  any  two  curves 
of  a  series  will  approach  a  certain  limit,  as  the  two  curves  approach 
coincidence.  Now,  if  we  suppose  the  parameter  to  vary  by  infinitesi- 
mal increments,  the  locus  of  the  ultimate  intersections  of  consecutive- 
curves  is  called  the  envelope  of  the  series. 

214 


ENVELOPES 


21  r> 


179.    The  Envelope  of  a  Series  of  Curves  is  Tangent  to  Every  Curve  of 
the  Series. 

P  0 


Suppose  L,  M,  X  to  bo  ;iny  three  curves  of  the  scries.  /'  is  tlic 
iutersectiou  of  J/  with  the  preceding  curve  L.  unci  Q  its  intersection 
with  the  following  curve  N. 

As  the  curves  approach  coincidence,  I*  and  Q  will  ultimately  be 
two  consecutive  points  of  the  envelope  and  of  the  curve  M.  Hence 
tlie  envelope  touches  M. 

Similarly,  it  may  be  shown  that  the  envelope  touches  any  other 
curve  of  the  series. 

180.    To  find  the  Equation  of  the  Envelope  of  a  Given  Series  of  Curves. 

IJefore  considering  the  yeneral  2)i()l)I('in  h't  us  take  the  foHowing 
special  example. 

Eequired  the  envelope  of  the  series 
of  straight  lines  represented  by 

,    ill 

y  =  «-^'  +  -, 
a 

a  being  the  variable  parameter. 

Let  the   equations   of   any    two   of 
these  lines  be 

,  in 

y  =  ax  +  -,  .     .     .     . 


and     y  =  (a  +  h)  x  + 


•  (1) 

•  (2) 


From  (1)  and  (2)  as  simultaneous 
equations,  we  can  find  the  intersec- 
tion of  the  two  lines.     Subtracting  (1)  from  (2) 


216  DIFFERENTIAL   CALCULUS 

km 


0  =  hx 


a(a  +  h) 


O^x ^L_ (3) 

a  (a  +  h) 


From  (3)  and  (1),  we  hav 


a(a  +  /i)  a(a+/i) 

which  are  the  coordinates  of  the  intersection. 

Now  if  we  suppose  h  to  approach  zero  in  (4),  we  have  for  the  nlti- 
mate  intersection  of  consecutive  lines 

,_m       _  2m 
a-    '         a 

By  eliminating  a  between  these  eqnations  we  have 

y-  =  4  mx, 

which,  being  independent  of  a,  is  the  equation  of  the  locus  of  the  in- 
tersection of  any  two  consecutive  lines,  that  is,  the  equation  of  the 
required  envelope. 

The  figure  shows  the  straight  lines,  and  the  envelope,  which  is  a 
parabola. 

181.   We  will  now  give  the  general  solution. 
Let  the  given  equation  be 

/(.T,  y,  a)  =  0, 

which,  by  varying  the  parameter  a,  represents  the  series  of  curves. 

To  find  the  intersection  of  any  two  curves  of  the  series,  we  com- 
bine 

f(x,y,a)  =  0, (1) 

and  f{x,  y,a-\-h)=Q (2) 


ENVELOPES  217 

From  (1)  and  (2),  we  have 

f{x,  y,a  +  h)  -f(x,  y,  a)  _^ 

and  it  is  evident  that  tlie  intersection  may  be  foiiiid  by  conibiniiiLr 
(1)  and  (3),  instead  of  (1)  and  (2). 

When  the  two  curves  approach  coincidence,  h  approaclies  zero, 
and  we  have,  by  Art.  15,  for  the  limit  of  equation  (o), 

£/(.T,2/,a)  =  0 (4) 

Thus  equations  (1)  and  (4)  determine  the  intersection  of  two  con- 
secutive curves.  By  eliminating  a  between  (1)  and  (4)  we  shall 
obtain  the  equation  of  the  locus  of  these  ultimate  intersections, 
which  is  the  equation  of  the  envelope. 

182.   Applying  this  method  to  the  preceding  example, 

a 
we  differentiate  with  respect  to  a,  and  obtain  for  (4)  Art.  181, 

cr 

Eliminating  a  between  these  equations  gives  the  equation  of  the 
envelope, 

y'-  =  4:mx,     as  found  in  Art.  180. 


183.   The  Evolute  of  a  Given  Curve  is  the  Envelope  of  its  Normals. 
This  is  indicated  by  the  figure  of  vVrt.  166,  and  the  i)r()positiun 
may  be  proved  by  the  method  of  Art  181,  as  follows  : 

The   general   equation  of  the  normal  at  the  point  (x',  y')  is  by 

(3),  Art.  148,  x-x'  +  '^'\{y-y')  =  0, (1) 

dx' 


218  DIFFERENTIAL   CALCULUS 

chi' 

in  which  the  variable  r)araineter  is  x',  the  quantities  y',  -^,  beiu" 

functions  of  x'.     Differentiating  (1)  with  respect  to  x',  we  have 

From    (1)    and  (2)  we  find   for   the   intersection   of   consecutive 
normals, 

y  =  y'  + 


dy' 
dx' 

[^-(r 

dx''-  *^ 

As  these  expressions  are  identical  with  the  coordinates  of  the 
centre  of  curvature  in  Art.  165,  it  follows  that  the  envelope  of  the 
normals  coincides  with  the  evolute. 

EXAMPLES 

1.  Find  the  envelope  of  the  series  of  straight  lines  represented  by 
y  —  2  mx  4-  m'^,     m  being  the  variable  parameter. 

Differentiating  the  given  equation  with  reference  to  m, 

0  =  2x  +  4m'''. 
Eliminating  m  between  the  two  equations,  we  have  for  the  envelope, 
ir)y^  +  27.);^  =  0. 

2.  Find  the  envelope  of  the  series  of  parabolas 

y-  =a(x  —  a),  a  being  the  variable  parameter.    Ans.    4:y-  =  X'. 

3.  Find  the  envelope  of  a  series  of  circles  whose  centres  are  on 
the  axis  of  X,  and  radii  proportional  to  (m  times)  their  distance 
from  the  origin.  A7is.    y-=:m-{^x^ -{-y'^). 


ENVELOPES  219 

4.  Find  the  evolute  of  tlie  parabola  ?/-  =  4ax-  accordiug  to  Art. 
183,  taking  the  equation  of  the  normal  in  the  form 

y  =  m (x  —  2 a)  —  avi'.  Ann.    'llaif  =  4 (x-  —  2 af. 

5.  Find  the  evolute  of  the  ellipse   •^"^^  +  ^^^  =  1,  taking  the  equation 
of  the  normal  in  the  form  '* 

by  =  ax  tan  cfy—  (a-  —  h'-)  sin  <ji, 
where  <j>  is  the  eccentric  angle. 

Arts.    {ax)i  +  {hy)^  =  (o-  -  U^ K 

6.  Find  the  envelope  of  the  straight  lines  represented  by 

.rcoso^  +  ?/sin;>^  =  a(cos2^)^,  •    ^Q- 
$  being  the  variable  parameter.  i^<-^^  0 

Ans.    {x- +  y-y- =  a- {ur  —  y^),     the  lemniscate. 

7.  Find  the  envelope  of  the  series  of  ellipses,  whose  axes  coincide 
and  whose  area  is  constant. 

The  equation  of  the  ellipses  is 

^+yj=i a) 

a-      h- 
a  and  h  being  variable  parameters,  subject  to  the  condition 

«^'  =  A-', (2) 

calling  the  constant  area  ttJc'. 

Substituting  in  (1)  the  value  of  b  from  (2), 

x^      a-y- 


(3) 


in  which  a  is  the  only  variable  parameter.     Differentiating  (3)  with 

respect  to  a,  we  have 

-i4+iff=o (4) 

Eliminating  a  between  (3)  and  (4),  we  have 
4  x-y-  =  k\ 


220  DIFFERENTIAL   CALCULUS 

Second  Solution.      Differentiate   (1),  regarding  both  a  and   b  as 
variable. 

^^  +  1^  =  0 (5) 

Differentiating  (2)  also,  we  have 

hda  +  adh^Q.     .     .■ (6) 

From  (5)  and  (6),  we  have 

t^yl (7) 

From  (7)  and  (1), 

t  =  t^l (8) 

a'     b'     2  ^  ^ 

Substitnting  (8)  in  (2), 

4  xY  =  JcK 

8.    Find  the  envelope  of  the  circles  whose  diameters  are  the  double 
ordinates  of  the  parabola  F  =  "^  ^^-  ^1"^-   V'  —  4^  c({,ci  +  x). 


X  ,  y 
when  a"  +  b"  =  k"- 


9.    Find  the  envelope  of  the  straight  lines      -  +--^  =  1, 

a      b 


Am.   a;"+' +  2/"+i  =  A;" 


10.  Find  the  envelope  of  the  ellipses       ^  +  ^  =  1, 

a-     ¥ 

when  a-\-b  ^k.  Ans.   x^  +  y^  =  k^. 

11.  Find  the  envelope  of  the  circles  passing  through  the  origin, 
whose  centres  are  on  the  parabola      ?/^  =  4  ax. 

Ans.    {x-{-2a)y-  +  a^  =  0. 


ENVKI.Ol'KS  '2-2\ 

12.    Find  the  envelope  of  circles  described  on  the  central  radii  of 
an  ellipse  as  diameters,  the  equation  of  the  ellipse  being 

—  +  — „  =  1.  Ans.    (ur  +  irf  =  a-ir  +  b-y-. 


13.  Find  the  envelope  of  the  ellipses  wliose  axes  coincide,  and 
such  that  the  distance  between  the  extremities  of  the  major  and 
minor  axes  is  constant  and  equal  to  k. 

Ans.   A  square  whose  sides  are  (.c  ±  y)'-  =  A;'*. 


INTEGRAL  CALCULUS 

CHAPTER   XX 
INTEGRATION.     STANDARD  FORMS 

184.  Definition  of  Integration.  The  operation  inverse  to  differ- 
entiation is  called  integration.  By  differentiation  we  find  the  dif- 
ferential of  a  given  function,  and  by  integration  we  find  the  function 
corresponding  to  a  given  differential.  This  function  is  called  the 
integral  of  the  differential. 

For  instance, 
since  2xdx  is  the  differential  of  ar, 

therefore  or  is  the  integral  of  2xdx. 

The  symbol  |  is  us&d  to  denote  the  integral  of  the  expression 
following  it. 


d(xr)=2xdx,   C2xdx  = 


It  is  evidently  the  samel.hing,  whether  we  considtT  this  integral 
as  the  function  whose  differential  is  2xdx,  or  tlie  function  whose 
derivative  is  2x. 

As  regards  notation,  liowever,  it  is  customary  to  write 


I  2xdx  =  .r,     and  not   |  2j;  =  x^. 


223 


224  INTEGRAL   CALCULUS 

In  other  words, 


/d 
is  the  inverse  of  d,  and  not  of  — 
dx 

Thus  the  general  definition  of    |  <fy(x)dx  is  that  function  whose 

differential  is  cf>(x)dx;  the  symbol    |    denoting  ''the  function  whose 

differential  is,"  in  the  same  way  that  the  inverse  symbol,  tan~^, 
denotes  "the  angle  whose  tangent  is." 

Integration  is  not  like  differentiation  a  direct  operation,  but  con- 
sists in  recognizing  the  given  expression  as  the  differential  of  a 
known  function,  or  in  reducing  it  to  a  form  where  such  recognition 
is  possible. 

185.    Elementary  Principles. 

(a)  It  is  evident  that  we  may  write 

C2  X  dx  =  a;-  +  2,  or    C2xdx  =  x\—  5, 

as  well  as  i  2xdx  =  x^', 

since  the  differential  of    x^  +  2,    as  well  as  of    05^  —  5    is    2x  dx. 

In  general  -   (2xdx  —  x-  +  C, 

wliere  O  denotes  an  arbitrary  constant  called  the  constant  of  integra- 
tion. 

Every  integral  in  its  most  general  form  includes  this  term, 
+  C. 

(b)  Since  d{ u  ±v  ±  to)  ^  du  ±  dv  ±  dw, 

it  follows  that 

I  (du  ±  dv  ±  dw)  =   I  du  ±  i  dv  ±  i  dw: 


INTEGRATIOX.     STAXDAKD    TOUMS  225 

That  is,  we  integrate  a  polyuumial  by  iutegialiiig  tlie  s-.'parate 
terms,  and  retaining  the  signs. 

(c)  Since  d(an)  =  adn, 

it  follows  that  j  adu  =  n  |  da. 

That  is,  a  constant  factor  may  be  transferred  from  one  side  of  lln- 
symbol   |   to  the  other,  without  affecting  the  integral. 

186.  Fundamental  Integrals.  Since  integration  is  the  inverse  oi 
differentiation,  to  integrate  any  given  function  we  must  reduce  it  to 
one  or  more  of  the  differentials  of  the  elementary  functions,  ex- 
pressed by  the  fundamental  formulae  of  the  Differential  Calculus. 
Corresponding  to  these  formuhe  we  may  write  a  list  of  integrals, 
which  may  be  regarded  as  fundamental,  and  to  which  all  integrals 
should,  if  possible,  be  ultimately  reduced.  We  shall  then  consider 
in  this  chapter  such  examples  as  are  integrable  by  these  formula;, 
either  directly,  or  after  some  simple  transformation. 


/ 


j  }i"da- 

:l0: 


II.  r* 

J   u 


du 


III.     I  a"du-.      "" 


J  log  a 

IV.     Ce^du  =  e". 

'       V.     I  cos  u  du  =  sin  u. 

VI.     j  sin  u  du  =  —  cos 


226  INTEGRAL   CALCULUS 

VII.  I  sec-  u  da  =  tan  u. 

■,  ■  VIII.  i  cose(i-udii  =  —  Gotu. 

IX.  I  sec  u  tan  u  du  =  sec  u. 

X.  I  cosec  u  cot  u  du  =  —  cosec  u. 

^,        XI.  I  tan  M  du  =  log  sec  w.  **'<y  Cos  U 

V      XIT.  j  cot  It  dw  =  log  sin  M. 

XI-II.  j  sec  w  da  =  log  (sec  ?<  +  tan  u)  =  log  tan  /^  E  -j_  !f  j. 

XIV.  I  cosec  u  du  =  log  (cosec  u  —  cot  w)  =  log  tan  - . 

XV.  I  ,  =  -  tan  ^  -,  or  = cot  '  -  • 

J  a-  +  a-      a  a  a  a 

VT7-T  r    flu  In      u  —  a  1   ,      a  — w 

.     XVI.  I  =  —  log ,  or  =  — log 

J  u-  —  or     2a        u+  a  2a        a  +  u 


[I.     f     ^. 

^  Va-  —  ur 


sin~'  — ,  or  =  —  cos~^  - 
a  a 


XVIII.     ^—^^^  =  log  («  +  ^W^^). 

XIX.     I  =-sec  i-,or=: ( 

J  u^ii?  -  a^      «  a  a 

f- 


XX. 


■  =  vers"'  -  • 
V2  aw  —  u-  ^ 


INTEGRATION.     STANDARD   FORMS  2l'7 

INTEGRALS  BY  I.  AND  II. 

187.    Proof  of  I.  and  II. 

To  derive  I., 
since  cZ(m"+^)  =  (n  +  l)^  du, 

therefore 

w"-"'  =  f{n  +  lyr  du  =  (n  +  l)Cu"  du,      by  (c),  Art.  18^. 


u"  di(  =  — 

n  + 1 

Formula  II.  follows  directly  from 


J ,  du 

d  log  u  =  ■ —  • 
u 

It  is  to  be  noticed  that  I.  applies  to  all  values  of  n  except  ?i  =  —1. 
For  this  value  it  gives 

r  -1  7        "" 

I  u  '  du  =  —  =00  . 

J  0 

Formula  II.  provides  for  this  failing  case  of  I. 

EXAM  PLES 
Integrate  the  following  expressions : 

1.    Cx*dx. 


S^ 


If  we  apply  I.,  calling  u  =^  x,  and  7i=4;  thendn=dx.      Then 
we  have 

Cai^dx  =  —-\-  C,  adding  the  constant  of  integration  C,  according  to 
J  5 

(a),  Art.  185. 


228  INTEGRAL   CALCULUS 

2.    C(x"  +  l)k'dx. 

If  we  apply  I.,  calling  m  =  ar  +  1,  and  n  =  -  ;    then  du  =  2xdx. 

Z 

We  must  then  introduce  a  factor  2  before  the  xdx,  and   conse- 
quently its  reciprocal  -  on  the  left  of   | 

C(x'  +  lyx  dx  =  I  C(x  2  + 1)  -  2  a;  dx,  by  (c),  Art.  185.    -^ 

2       3  3        * 

2 

2     r(a;^  -  a-)c7a;  ^  1  r(3.g"- 3a-)r?a; 
J    a^  —  3  a-a;      3  J      a^  —  3  a^x 

=  llog  ix"'  -  3  fr.r)  =  log  (a;'^  -  3  a-xf  +  C. 
o 

By  introducing  the  factor  3,  we  make  the  numerator  the  differ- 
ential of  the  denominator,  and  then  apply  II. 

4.    C(2x^-^x''+12x?-^)dx=—  -  ^'+3a;*-3a;+(7. 
J  5  7 

J\         ^V  x^     x)  5  2  a;^  ^     ^ 

6.  r  (.r'  -  2)'lr  dx  =  -^^i-"  -  '^'^  +  2  a,-«  -  2  a;"  +  C. 

7.  |'(.x.=  -2)^BfU-  =  ^-^''-^^-^4-C. 


001 » 


INTEGRATION.      STAXDAKI)    FORMS 


11.    j  L^3  +  a;  iW/x^'^z-f  ^11-+ 6.i-  +  12a-^-3a;-s4-C'. 

J  ^i  7  4 

14.  J(.r^  +  l)^x.-^da;=(^':!J^"  +  a 

15.  (k{a^^li)^xdx.  16.     ("((M- +  70"f'-^"- 
^17.     r(aar'  +  &)"a;-(?a.'.  18.     pa./-" +  /0''-k""' ('a?. 

-•    21.   J(a;-»-iy«^.  22.  )  Jo*+logOf 


12. 


^    13 


230  INTEGRAL   CALCULUS 

25.     f^^A z  +  ^r^le^dx-       26.     fsin^^  cos  ^  d^. 

J  |_(e-'  +  2)2      e-*  — 3J  J 

27.     r(e2«  +  sin  26)  (e^^  +  cos  2^)d^. 

28. ,    I  tan^  x  sec-  x  dx.  29.     |  sec^a;  tan  x  dx. 

30.  C{sm^O  +  cos"^)  sin  0  cos  ^  dO. 

31.  ("(sec  ^  +  tan  ^y^sec  6  dd. 

32.  r(sin  </)  +  cos  </))"(sin2  </>  —  cos''  <^)d(^.     33.     \{a'  +  5)3  a*  d«. 


34. 


/^sin~^  X  dx  ^  gc      C dx 

^    Vi~^^-  '   «>'  (1  + '^•■)  tan-i; 


A  rational  fraction,  whose  denominator  is  of  the  first  degree,  may 
be  integrated  directly  or  after  being  reduced  to  a  mixed  quantity. 


""■  /lfT-3''^=T-T'°^(-^+^)+''- 


38. 


■f-^TS^'-=f-f-T-S-^(^^-^>+'^- 


INTEGRATION.     STANDARD    I-OIIMS  231 

Oft      Cax  +  h  J        ax  ,  b- —  a- •,       ,, 


^^  ^^^^f  '''''  =  f + T  "^  "'''*  +  - "'  ^^"  ^^^  -  "^  +  ^^• 


41 .  f^^^r-f-  ^^  =  ^  +  2 ax-  bx  +  (a  -  bf  log  (a;  +  6)  +  C. 

42.  r(^±_^'  (^a;  =  ^'  +  2  aar'  +  7a'x  +  S  a"  log  (x  -  a)  +  C. 

INTEGRALS   BY   III.   AND   IV. 

188.    Proof  of  III.  and  IV.     These  are  evidently  obtained  directly 
from  the  corresponding  formuliB  of  differentiation. 

EXAMPLES 


232  'INTEGRAL   CALCULUS 

5.     r(e^+'' -  6"^+*)  da.\  6.     C(e''"''cosx-a''°'-'sm2x)dx. 

\  tN  Cfe^%  +  -\ dx.  8.     Ae'"" « sec  ^  -  6^^"= « tan  6) sec 6 dd. 

re''"'^sm-xdx         .^      T  xz-x^         T/  j.-'nx  7           a^^^""      ,   ry 
9.     I 10.     I  a^'b-'dx—  I  (ab-Ydx  =  — ■  + C. 

log(a'"6^) 
J        e^  e^\21oga-l      21og6-ly' 

/  J^  ^  2  log  6      log  (a6)      2  log  a 

INTEGRALS   BY   V.  — XIV. 

189.   Proof  of  V.  —  XI V.     It  is  evident  that  V.—X.  are  obtained 
directly  from  the  corresponding  formnlae  of  differentiation. 
To  derive  XI.  and  XII., 

/tan  u  da  =  —  i =  —  log  cos  u  =  log  sec  u. 
J       cos  u 

\ 

/,      ,         /^cos  u  du     T        • 
cot  u  du  =  I  =  log  sm  21. 
J     sm  It 

To  derive  XIII.  and  XIV., 

/fj    —  /*sec  u  (tan  u  +  sec  tt)  dn  _  /^sec  U  t^u  du  +  sec*7<  du 
J  sec  u  +  tan  u  J  sec  u  +  ta^M 

=  log  (sec  u  +  tan  rt) . 

/,•        Tcosec  M  (  —  cot  w  +  cosec  u)  du 
cosec  u  du  =  I  5^ — ^ — 
J               cosec  M—  cotw 

=  log  (cosec  u  —  cot  u). 


I 

INTEGRATION.  STANDARD  FORMS        233 

By  Trigonometry, 

2  sin-  - 
1  —  cos  u  2         .      u 

cosec  u  —  cot  u  =  — -. = =  tan  -• 

sin «         o   .    u       a  2 

\  2  sin -cos - 

\  w       2 

^  If  we  substitute  in  this,  -  ■j-,u  for  v, 

we  have  sec  u  +  tan  w  =  tan  (  -  +  ^' 

Thus  we  obtain  the  second  forms  of  XIII.  and  XIV. 

1      r/  •    o      ,          -           ■    ^\  1            cos.S.r  ,  sin  5; 
1.     I  (  sin  3  X  +  cos  ox—  sin  -   dx  = ^-  — ;;;— 


2.  I  (  sm  —^ \-  cos  —^—    dx  =  —  m  cos  —^—-  +  n  sm  ^—  +  C. 

J  \         m  n    J  m  n 

3.  p  +  sin  >».r  ^^ ^.  ^  1_  ^^^^^  ^^^^  _^  gg^  ^^^_^^  _^  ^^ 
»/      cos-  »l.«  ^  ?u 

4.  I  (sec  5x  —  tan o .^•)  sec 5 x dx.       5.    i  (sec 20  +  tan  2 ^) dd. 

JUos-^      sin^y  J 

.    8.    r-^^^^l^cZ-T  =  cos  .T- 2  log  (1+ cos  a-)  +  C. 
f  »/     sin  a* 

;    9.    fX^dx.  ,      10.     f^^d. 

I  J   cos-.T  I  -y    sin- A' 

jj      r sec<^r/c^ =  llogr</tan<^  +  ?>)  +  C'. 

J  a  sin  <^  +  6  cos  <^      a 


7.        (sin  ^- vers  ^)-(/^. 


234  INTEGRAL   CALCULUS 

12.  I  (tan  X  —  cot  x  +  1)-  dx  =  tan  x  +  cot  a;  —  3  x—  2  log  sin  2  a;  +  C. 

13.  I  (sec 2x-{-  tan  2x—  cot  2 x)- dx  —  tan 2 «  +  sec 2x  —  cot 2 x 
«/  2 

— log  tan  «  —  4  x  +  C 

14.  I  (sec  <^  +  cosec  <^  —  1)-  d^  =  <^  +  tan  c^  —  cot  ^ 

+  21ogl±^^+(7. 
1  +  sin  </) 

The  following  may  be  integrated  after  trigonometric  transforma- 
tion. 

15.  Csm^xdx  =  ^-'^^^+a 
J  2  4 

16.  Cco?xdx^-  +  ^^^^+a 

J       •-  2  4: 

17.  Cyers'xdx.  18.     Csm^xcos''xdx  =  ^-^^^^+ C. 

19.  I  sm  mQ  sin  nO  dd  =  —-i ' ;— > — — f-  +  G. 

J  2  (m  -  n)  2  (m  +  n) 

20.  I  cos  mO  cos  nO  dd  =  —-) :f-  +  —pr^ — —f-  +  O. 

J  2(771  — n)  2(m  +  ?i) 

ni      r  ■       f\  f\  jn  COS  (m  —  n)Q     cos  (m  4-  »)^  .   n 

21.  I  sm  m^  cos  nO  dd  = ~ — ^ — f-  +  C 

J  2(m— ?i)  2  (m+«) 

oo     /^       r  o     J        sin .']  x  ,  sin  7  .t  ,   ^ 

22.  I  cos  5  X  cos  2  .t;  da;  =  ■ — 1 — 1-  C. 

J  6  14 


; 


INTEGRATION.     STANDARD    FORMS  235 

23.  Jsin  {3x  +  2)  cos  (4  x  +  3)  dx  =  ^"'^(f+^)  _  co8(7x  +  5)  _^  ^, 

^  24  16  8       ^ 

25.    f'-^cie.  26.    r^"4^rf^  =  2sin^  +  2sin3_^ 

27.    f— ^  =  tan^-sec^  +  a  28.    f-^.       , 

J  1  +  sill  ^  J  vers  ^ 

oo     ri  —  sin.T  J  .^        ,  ^. 

«69.    I dx  —  —  cosec  x  —  cot  a;  —  log  vers  x  +  C. 

J     versa:  ^ 


Qft     r  /5 r—  ,         i^  COS  X  dx  ^    , 

30-    I  Vl+smxdx=  I     ,         .       =-2Vl-sina;+  O. 
^  ^  VI—  sill  a; 

J  Jsm^  +  cos^      V2  Us; 


INTEGRALS   BY   XV.  — XX. 

190.    Proof  Of  XV.  — XX. 

To  derive  XV., 

du  ^j 


—  _  ran   • 


To  derive  XVII., 

du 


•^  V  ft"  —  w-     *^      / 1  _  H!  ^ 


236  INTECxRAL   CALCULUS 

To  derive  XIX., 

du 


/clu  1  r  a  1  ,u 
—  =  _  I —  _  sec~  — * 
M Vm''  —  a-      "'^  w    /«'  _  1      ^  <^ 


To  derive  XX., 

dti 


/du         __  r        a 
■\/2au  —  u^     ^      I  11     u~ 
a     a^ 


vers"^  -• 
a 


Since  tan-^  _  =  '^  -  cot-^  -, 

a      2  a 


it  is  evident  that 


a        \ 


Hence  either  expression  may  be  used  as  the  integral  in  XV. 

In  the  same  way  we  obtain  the  second  forms  of  XVII.  and  XIX.. 

The  formuhe  XVI.  and  XVIII.  are  inserted  in  the  list  of  integrals, 
because  their  forms  are  similar  to  XV.  and  XVII.,  respectively,  with 
different  signs. 


To  derive  XVI., 


u~  —  a-     J  a\u  —  a     ?t  +  a 

hence 

r    du     ^  1    rf  du  du  \ 

J  u^  —  a^     2  a  J  \w  —  a     u-\-aJ 


^^[,og(.„-„)-,og(,.  +  a)]=J-,og'i^«. 


INTEGRATION.     STANDARD   FORMS  -231 

Or  we  may  integrate  thus  : 

r    du     ^  1    rf~du       du   \ 
J  u^  —  d-     2aJ  \a  —  xi,     a  +  u) 

=  A  [log  (a  -  u)  -  log  (a  +  «)]  =  J-  log  ^^^ . 
-i «  2a        a-ifu 

To  derive  XVIII., 


assume  Vm^  ±  a-  =  2,    a  new  variable. 

Then  w^  ±  a'  =  g-, 

2udn  =  22;f72; ; 
(i?<      dz      dn  4-  (fz 


therefore 


Z  U  XL+Z 


Hence  f  ^^  =  nin±dz  ^  j     ^^^  ^ 

J    z       J     u-\-z 

/du  


that  is 


EXAMPLES 


4 
1^ 


1.     f-^?^  =  ltan-^'+C'.     2.    f-^^  =  Ilog2^:=:^+C'. 

3.     I     ^  =  -  siu  '  -—  +  C. 

-^^  V4-25a^     «  ^ 

r        ^^     -=llog(5a;+V25x--4)  +  a 
-^  V25X---4     5 


5.     f '^ =  -L\oir(xV5+V5x''  +  l)  +  a 

•^  Voar'  +  l      V5 


238 


INTEGRAL   CALCULUS 


J3-I2x'  12     '"2a;-l^ 

7   c  ^^y  8   r  ^^'" 

■   J  I2/  +  3'  ■  J  12i(;2-3' 

0.     f       ^^^       .  11.     f       ^^^_ 

-^  V3  i^  -  2  -^  V2^3  a;' 


/^^•/j 


fZrr  1      ^_i  3  X 


a 


V9a;--4      ^ 


14. 


/ 


-y/mx 


dx                      -1^  ^  ,    n 
:  =  vers  '- \-  G. 


itr      C         dx  1       _i  ax  ,  ^ 

15.     I z==  =  -  sec  ^  -  -  +  G. 


ic    r     dx        1      _i  8 x-    ^ 

'^  Via;  — 4x-      ^  ' 


^'•/- 

»/  ^ 


dx 


=  sin~^  log  Va;  +  G. 


a;V4  —  (logx)' 


»•/: 


dic 


12 


3  0^-5 
dx 


r     dx 


tan-'  -^  +  a 


19. 


r-^d.  =  llog(4ar'_5)+-^log?AZLV_5^C 
j4ic^-5  8  4V5        2.7:4- A/f) 


20.     f^^-^  dx  =  -3V9-x'-2  siu-i  ^  ^  c'^ 


INTEGRATION.     .srAXDARD   FORMS  239 

21 .  f  ^_±^  civ  =  V^?T4  +  3  log  (.T  +  a/.'?T4)  +  a 

22.  f  ^^^  da;  =  ^V3ar-9-J:^log(.TV3  +  V3x"-9)+a 

J  V3^'-9        -^  Va 

23.  f  .,  _    ^^V       ,    ^ltan--^^^iHli+(7. 
»/  a-  sin-  ^  +  6"  cos-  B      ah  b 

24.  f      '^^        ^J_tau-^tan^  +  a 
J  1  +  cos-  <^      V2  V2 


sin  6d0  „  „  „_i  A'os  i 

+  4  sin- 1 


25.     f ^"i^" =  cos-^  1^.1!^]^  a 


26.  C  sm2o:dx  =  _  1  log(3  cos2.r+ V9cos-'2a;-4)+C. 
•^  V5  cos-  2  05— 4  sin^  2  a;         ^ 

27.  p_^±^'  dx-  =  l  log  (e'-^  +  a)  +  -1^  tan-i  -^  +  C. 
J  e^  +  a  2  Va  Va 

The  same  fornuilse  may  be  applied  to  integrals  involving 
x^-{.ax  +  b  or  —x^  +  ax  +  b,  by  completing  the  square  with  the 
terms  containing  x.     Thus, 

Jx^  +  6x+13     J(a;  +  3)^'  +  4     2  2 

29.     f  ^^  =  r  ^^  =1  sin-  ^^  -f  a 

^  V8  +  4  X  -  4  x^     -^  Vy  -  (2  X  -  1/     ^  "^ 


30.     f  ^^        —  =  -l-log(3a;-2+V9ar'-12g4-6)+C. 

^  V3a.-2-4a;  +  2      V3 


240  INTEGRAL   CALCULUS 

31.  I  — =  — r=  tan-^ ^  +  C. 

J  2x--3x  +  5      vai  V31 

32.  I  — ^ ,  wheu  a  =  4  ;  when  a  =  6  ;  when  a  =  8. 

J  X-  —  ax  +  9     • 


33.  r *^ -,  =  -J-tan-^-^-  +  ^+-^+a 

J  (x  +  a)-  +  (a;  +  ^)-      a  —  ^  a  —  b^ 

34.  r  "-^ =  ±  sin-i "•"!"  +  a 


35.     f  ^-^  =-J-log^^+J  +  (7. 

^W(x  +  a)(x  +  b)      «-^        •^•  +  a 


oc       /^  dx  ■     -,2  x  —  a  —  b  ,   ^ 

36.     I  —  =  sm-i 1-  C. 

J  •x/i' 7-  _  r/Ai' 7i  —  '^\  a  —  b 


V{x-a){b-x)  «  — & 

2 

-^-^ t 


37.     C, '^ = 2 tan-^g-^--+«  +  ^l-^+C. 


38. 


/^^  

/,    ^  '        ===21og(V..  +  a  +  V.T  +  6)  +  G 
V(a;  +  a)(.T  +  6) 


CHAPTER    XX T 


SIMPLE  APPLICATIONS   OF   INTEGRATION. 
TEGRATION 


CONSTANT    OF   IN- 


Before  coutinuing  the  integration  of  functions,  we  will  consider 
the  relation  of  integration  to  the  determination  of  the  area  bounded 
by  a  given  curve,  and  show  how  the  constant  of  integration  may  be 
determined. 

191.  Derivative  of  an  Area.  Let  ?/=/(. f)  be  the  equation  of  a  given 
curve  OPj.  Suppose  a  point  to  move  along  the  curve  starting  from 
/oj  and  let  X,  y,  be  the  coordi-  y 
nates  of  any  position  P. 

At  the  same  time  the  ordi- 
nate of  the  moving  point 
starts  from  the  position 
P^)M^),  and  sweeps  over  or 
generates  a  certain  area. 
When  the  point  has  moved 
to  P,  this  area  is  P^Mf^MP. 
Denote  this  area  by  A. 

^  is  a  function  of  x,  and  it 
will  now  be  proved  that  its  derivative  with  respect  to  x  is  equal  to  >j. 

Give  to  X  the  increment  A.x'  =  MX. 


^^_^ 

'P^ 

J^ 

F" 

^^ 

p„ 

y^ 

A 

y 

A  A 

A  a- 

M     N 


M,  X 


Then 


Hence 


A^  =  PMNQ. 

A.4  >  y  Aa;,  and   A.l  <  (//  -f  A//)  A.r, 

A^ 

Ax 

(lA 


>y, 


and  ^^'   <  y  +  Ay. 

A.C 


=  Lim^^^ 


AJ 


=  .'/• 


dx  A.C 

In  case  the  curve  descends  from  P  to  Q,  the  above  inequalities 
will  be  reversed,  but  the  result  will  be  the  same. 

241 


242  INTEGRAL   CALCULUS 

192.  Area  of  Curve.  Let  it  be  required  to  find  the  area,  Po3foMiPy, 
between  the  curve,  the  axis  of  X,  and  the  two  ordinates  PqMq  and 
P,3f,. 

Let  03fo  =  a,  and  OJfi  =  b. 

From  the  preceding  article  ~  =y. 


Hence  A=  i  ydx=  (  f{x)dx. 

Let  ff(x)clx  =  F{x)  +  C; 


then  A  =  F(x)  +  C. (1) 

To  determine  C,  we  have  the  condition  that  A  begins  when  x=a; 
that  is,  A  =  0  when  x  =  a. 

Hence  0  =  F{a)  +  C,     C=-  F(a). 

Substituting  in  (1),  A  =  F(x)  -  F  (a)  =  P^M^MP.     ....     (2) 

It  is  to  be  noticed  that  C  is  determined  by  the  initial  value  a  of  x, 
corresponding  to  the  initial  ordinate  PoMn. 
If  now  we  let  x  —  b  in  (2),  we  have 

A  =  F{b)  -  F(a)  =  P,M,M,P,. 
For  example,  let  the  given  curve  be  the  parabola  y~  =  x. 
Then  A=  Cydx=Cxiclx='^-\-a       ...     (3) 

To  determine  C,       A  =  0  when  x  =  a. 

3  3 


; 


SIMPLE   APPLICATIONS   OF   INTEGRATION  243 

Substituting  in  (3), 

A=--f--'^=l\M,MP. (4) 

To  find  P^M^M.P,,  let  x=  h  in  (4). 

2  62      2  a^ 


3  3 


EXAMPLES 
1.    In  tlae  curve  of  Ex.   1,  p.  24,  show  tiiat  P^^OM^P^  = 


Also  PiJI^MP,  =  — 

12 


"      2.    Find   the   area  included  between    the    equilateral    hyperbola 
2xy  =  or,  the  axis  of  X,  and  two  ordi nates  x  =  a,  x  =  2  a. 

Ans.     a-  log  V2'. 

(/  i   3.   Find  the  area  included  between  the  witch  of  Agnesi  (Art.  126), 
the  axes  of  X  and  Y,  and  the  ordinate  x  =  2  a.  Ans.     -mr. 

/^ -4.    Find   the   area   included   between    the    catenary    (Art.    128), 
the  axis  of  X,  and  the  ordinates  x  —  a,  x  =  2a. 

Ans.     —  («-  — e  +  e~'— e~^, 

(/-    5.   Find  the  area  of  one  arch  of  y  =  sin  x.  Ana.     2. 

^''    B.    Find   the    area   included    between   the   parabola  x^+y-  =  a^ 

a* 
(Art.  129),  and  the  axes  of  X  and  Y.  Ans.     -. 

'         7.    Find  the    area   included   between    the    seinicubical    jjarabola 
at/-  =  3^  (Art.  130),  the  axis  of  Y,  and  two  abscissas.  ?/  =  S  a,  y  =  27  d. 

Ana.      -—  a; 
6 


244 


INTEGRAL   CALCULUS 


Conversely,  instead  of  determining  the  area  from  the  integral,  we 
may  find  the  integral  from  the  area,  when  it  can  be  obtained  geo- 
metrically from  the  figure.     For  example  : 

8.   Find    j  Va"  —  a?  dx,  by  means  of  the  curve  y  =  Va'  —  «",  circle 
about  0,  radius  a. 
A  =  BOMP  =  OMP+  BOP 


=  v,^y  +  o  '^  =  o-'*^' ^ '*'  -  •^'  +  ~  •'^^ 


a-    ■      ,  X 

sin-'-- 

a 


If  the  initial  ordinate,  instead  of  OB, 
had  been  some  other  ordinate,  we  should 
liave  had 


A  =  '  -\/a-  —  .1"^  +  ~  sin"'-  +  C,  where  C  is  independent  of  x. 
2  2  a 

Hence  ^1  =  j  ?/  (Lc=  j  Vtr—  -/c'-clx  —  -  Va-  —  x--\ sin~'  ~  +  C. 

J  J 2 : 2      __«____ 

9.   Find    j  (.")  X  +  2)f?.i;,  by  means  of  the  line  y  =  ox  +  2. 
10.    Find    I  V2  ax  —  x-  dx,  by  means  of  the 

/v. 


ciirve  y  =  •\/2  a.^•  —  a^. 


Am 


ax  —  X-  dx= 


193.    Other  Illustrations.     In  order  to  further  illustrate  the  deter- 
•fc.  mination  of  the  constant  of  integration,  we  will  work  three  examples, 
involving  geometrical  or  physical  properties. 

Ex.  1.  Determine  the  equation  of  a  curve  through  the  point  (4,  3), 
at  every  point  of  which  the  slope  of  the  tangent  is  equal  to  the  recip- 
rocal of  twice  the  ordinate  of  the  point  of  contact. 


yi.Ml'LK    APPLICATIONS   OF    IXTE(; RATION 

By  the  hypothesis 


from  whieli 
Integrating, 


chi  ^  J_ 
dx      '2  If' 


2yd>i  =  dx. 
if  =  X  +  C. 


245 


(1) 


This  equation  represents  a  series  of  parabohis  whose  axes  coincide 
with  the  axis  of  x. 

If  now  we  impose  the  additional  (•()ndition  that  the  curve  must 
pass  through  the  point  (4,  o),  its  coordinates  must  satisfy  equation 

^^^''""^  9^4  + C,     C'  =  5.    . 


The  equation  of  the  particular  curve  is  therefore 

y-  =  .i;  +  5. 

Ex.  2.  A  body  starting  from  rest,  with  a  given  initial  velocity  r,„ 
moves  with  a  constant  acceleration  (j.  Find  the  space  passed  over 
in  any  time. 

In  Art.  19,  acceleration  =  —  • 
'  dt 


Here 
Integrating, 


dv  ,  ,, 

_  =  r/,  dc  =  (J  dt. 
dt 

v  =  gt-\-C.    1 


From  the  conditions  of  the  example,  r=  /•„  when  /  =  0;  thrndnn' 


246 

INTEGRAL   CALCULUJ; 

Hence 

V  =  gt  +  Vq. 

Since 

v=-  (Ai-t.  18), 
dt                ' 

Integrating, 

s  =  lgt-  +  i\,t+a 

From  the  conditions  of  the  example,  s  =  0  when  ^  =  0;  therefore 
(7=0,   and   s  =  ~ gf" -{- v^^t   is  the  complete  solution. 


Ex.  3.  A  body  is  projected  at  an  angle  a  with  the  horizon,  and 
with  a  velocity  i\,.     Find  the  equation  of  its  path. 

Kepresent  the  horizontal  and  verticail  components  of  the  velocity 
by  Wj  and  v^  respectively.  Then,  since  gravity  is  the  only  force  act- 
ing on  the  body,  we  have 


^  =  0,    and 
dt 

dt          •' 

Integrating, 

v.=  C, 

^V  =  -9^+  C- 

When  t  =  0, 

i\  =  Vq  cos  a, 

Vy  =  i\,  sin  a. 

Hence 

V^  =  I'n  cos  «, 

v^  —  —  gt  +  i\^ siu  a; 

at  is. 

dx 

-~-  =  ro  cos  a, 
dt 

^  =  -  r/^  +  r„  sin  a. 
dt 

I 


I  \         Intesiratino-.      x  =  vJ  cos  «  +  C.  V  v  = 


\ 


Integrating,      x  =  r\f  cos  «  +  C^  V         y  =  ^  -  gf  -f-  r,/  sin  a  +  C". 
When  ^  =  0,      x  and  ?/,  and  therefore  C  and  C",  are  zero. 

Hence  x  =  v^t  cos  a,  and     y  z=  —  -gf  -\-  v^t  sin  a. 

2i 

Eliminating  t  between  these  equations,  we  have  as  the  equation  of 
the  path  of  the  projectile, 

?/=.i'tan«  •'■ 


i\-  cos-  a 


This  evidently  represents  a  parabola  whose  axis  is  parallel  to  the 
axis  of  Y. 


SLMl'Li:    AI'I'LICATIOXS   OF    I XTIICILVTIOX  247 

EXAMPLES 

4.  Find  the  equation  of  the  curve  whose  subnormal  (Art.  14(i) 
has  the  constant  value  4,  and  which  passes  through  the  point  (1,  4). 

Alls,   if  =  8x4-8. 
I 

5.  Find  the  equation  of  the  curve  whose  subtangent  (Art.  14G) 
is  twice  the  abscissa  of  the  point  of  contact,  and  which  passes  through 
the  point  (2,  1).  j^ns.    x  =  2  if. 

6.  The  slope  of  the  tangent  to  a  curve  at  any  point  is '- ,  and 

the  curve  passes  through  the  point  (3,  2).     Find  its  ecpiation. 

.l)*.s.   4  .r -f  9  ?/- =  72. 

7.  Find  the  equation  of  the  curve  wliose  polar  subtangent  (Art. 
153)  is  3  times  the  length  of  the  corresponding  radius  vector,  and 

which  passes  through  the  point  (2,  0).  Ans.   r  =  2e''\ 

8.  Find  the  equation  of  the  curve  whose  polar  s\d)normal  (Art. 
153)  is  3  times  the  length  of  the  corresponding  radius  vector,  and 
which  passes  through  the  point  (2,  0).  ^,j5,    ,,  _2e*^. 


9.    Find  the  equation  of  a  curve  through  the  point  (  3,  -  j,  in  which 

the  angle  between  the  radius  vector  and  the  tangent  is   half  the 
vectorial  angle.  '    '  Ans.    /•  =  (;(! -cos  ^), 


10.  A  balloon  is  ascending  with  a  velocity  of  20  miles  an  hour. 
A  stone  dropped  from  the  balloon  reaches  the  ground  in  0  seconds. 
Find  the  height  of  the  balloon  when  the  stone  is  dropped, 

Alts.    400  ft. 

11.  If  a  particle  moves  so  that  its  velocities  parallel  to  tlie  axes 
of  X  and  Y  are  ki/  and  kx  respectively,  prove  that  its  path  is  an 
equilateral  hyperbola. 


248  INTEGRAL   CALCULUS 

12.  A  body  starts  fmm  tlie  origin  of  coordinates,  and  in  t  seconds 
its  velocity  parallel  to  the  axis  of  X  is  6 1,  and  its  velocity  parallel 
to  the  axis  of  I"  is  ?)f  —  o.  Eind  (a)  the  distances  traversed  parallel 
to  each  axis  in  t  seconds ;  (b)  the  distance  traversed  along  the  path  ; 
(c)  the  equation  of  the  path. 

Ans.    (a)        x  =  ot-;  y  —  f  —  'd  t. 

13.  If  a  body,  projected  from  the  top  of  a  tower  at  an  angle  of 
45°  above  the  horizontal  plane,  falls  in  5  seconds  at  a  distance  from 
the  bottom  of  the  tower  equal  to  its  height;  find  the  height  of  the 
tower  (g  =  32).  Ans.   200  ft. 

14.  When  the  brakes  are  put  on  a  train,  its  velocity  suffers  a  con- 
ctant  retardation.  If  the  brakes  will  bring  to  a  dead  stop  in  2  min- 
utes a  certain  train  running  30  miles  an  hour,  how  far  from  a  station 
should  the  brakes  be  applied,  if  the  train  is  to  stop  at  the  station  ? 

Ans.   Half  a  mile. 


CHAPTER    XXir 
INTEGRATION    OF    RATIONAL   FRACTIONS 

194.  Formulae  for  Integration  of  Rational  Functions.  On  examin- 
ing the  fundamental  integrals  in  Art.  18G,  it  will  be  seen  that  only 
four  apply  to  the  integration  of  rational  algebraic  functions,  I.,  II., 
XY.,  and  XVI. ;  and  of  these  only  I.,  II.,  and  XV.  are  independt^it, 
since  XVI.  depends  directly  upon  II. 

It  will  be  shown  in  this  chapter  that  by  these  three  formula?  any 
rational  function  can  be  integrated.  The  integration  of  a  rational 
polynomial  has  been  explained  in  Chapter  XX.  "We  will  now  con- 
sider the  integration  of  rational  fractions. 

195.  Preliminary  Operation.  If  the  degree  of  the  numerator  is 
equal  to,  or  greater  than,  that  of  the  denominator,  the  fraction 
should  be  reduced  to  a  mixed  quantity,  by  dividing  the  numerator 
by  the  denominator. 

For  example, 

.r'  +  l  x'  +  l  ' 

2^- ox*  +  1  ^  o ,^.  _  3  _^  -2ar'^  +  3  3f'-f  1  ^ 
X*  +  X-  '  X*  -t-  X- 

The  degree  of  the  numerator  of  the  new  fraction  will  be  less  than 
that  of  the  denominator. 

The  entire  part  of  the  mixed  (piantity  is  readily  intcgnible,  and 
thus  the  integration  iji  any  rational  fraction  is  made  to  depend  upon 
the  integration  of  one  whose  numei'ator  is  of  a  lower  degree  than  the 
denominator. 

24'J 


2.30  INTEGRxVL   CALCULUS 

196.  Partial  Fractions.  A  rational  fi-actiou  is  integrated  by  de- 
composing it  into  partial  fractions,  whose  denominators  are  the 
factors  of  the  original  denominator.  The  complete  discussion  of 
Partial  Fractions  belongs  to  Algebra.  .  We  shall  only  consider  here 
the  form  of  these  partial  fractions  and  the  processes  of  determining 
them. 

Factors  of  the  Denominator.  It  is  shown  by  the  Theory  of  Equa- 
tions that  a  polynomial  of  the  ?(th  degree,  with  respect  to  x,  may  be 
resolved  into  n  factors  of  the  first  degree, 

(x  —  a^)(x  —  a_,)(,«  —  ag)  •••(.«  —  «„). 

These  factors  are  real  or  imaginary,  but  the  imaginary  factors 
occur  in  pairs,  of  the  form 

x  —  a  +  b^  —  I,    and   x—  a  —  h^  —1, 

whose  product  is  (.r  —  a)"  +  h-,  a  real  factor  of  the  second  degree. 

It  follows  that  any  polynomial  may  be  resolved  into  real  factors 
of  the  first  or  second  degree,  and  only  such  factors  will  be  considered 
in  the  denominators  of  fractions. 

There  are  four  cases  to  be  considered. 

First.  Where  the  denominator  contains  factors  of  the  first  degree 
only,  each  of  which  occurs  but  once. 

Second.  Where  the  denominator  contains  factors  of  the  first 
degree  only,  some  of  which  are  repeated. 

TJiird.  Where  the  denominator  contains  factors  of  the  second 
degree,  each  of  which  occurs  but  once. 

Fourth.  Where  the  denominator  contains  factors  of  the  second 
degree,  some  of  which  are  repeated. 

197.  Case  I.  Factors  of  the  Denominator  all  of  the  First  Degree, 
and  none  repeated. 

The  given  fraction  may  be  decomposed  into  partial  fractions,  as    I 
shown  by  the  following  example, 


J      ar'  —  4  a; 


1 


INTEGRATION    OF   RATIONAL   FRACTIONS  261 

Assume 

x'-^.x         {x-2)(x  +  'I)x     x-'2^  x  +  2^  X       '     '     ^^ 
where  A,  B,  C  are  unknown  constants. 

■    Clearing  (1)  of  fractions, 

x^  +  6x-S  =  Ax{x  +  2)+Bx{x-2)+C(x-2)(x  +  2)      ...     (2) 
^(A  +  B+  C)  .r  +  2  (.1  -  B)  x  -  4  C. 

Equating  the  coefficients  of  like  jiowers  of  x  in  the  two  mem- 
bers of  the  equation,  according  to  the  method  of  Undetermined 
Coefficients,  we  have 

^1+B+C'=1,     2(A-B)  =  (>,     -4C==-8. 

whence  A^l,     B  =  -2,     C=2. 

a;2  +  r>.K-8         1  2,2 

x^  —  4:X         x  —  2      x-\-2      X 

and  f'^  (^x-S  ^^^  ^  ^^^^  ^_^.  _  2)  _  2  log  (.^•  +  2)  +  2  log  a? 

^        ,r(x-2)^ 
{x+2y 

The  following  is  a  shorter  method  of  finding  A,  B,  C: 
Suppose  the  denominator  of  the  given  fraction  to  contain  the  flic- 
tor  X  —  a,  not  repeated.     Then  the  fraction  may  be  expressed  as 

/(■>:)         ^     A      ^   ^p(x) 
{x  —  a)  <f>  i^x)      X  —  a      </)(.e) 

Hence  .ZM  =  .1 +(.._  c^--^^^. 

This  being  an  identical  equation  is  true  for  all  values  of  x. 


252  INTEGRAL   CALCULUS 

If  we  put  x  =  a,  we  have  .1=  •' W,  since  by  hypothesis  cf){x)  does 

not  vanish  when  x  =  a.  ^^'"^^ 

Thus  we  have  the  following  rule  : 

To  find.  y=l,  the  numerator  of  the  partial  fraction  -,  putx  =  a 

x  —  a 

in  the  given  fraction,  omitting  the  factor  x  —  a  itself. 

For  example,  having  written  equation  (1),  we  find  A  by  substitut- 

inga;  =  2in  the  given  fraction  — ^ — ^!-— ^ '■ — ,  omitting  the  factor 

o        mu-         •  (X  —  2)(X  +  '2)X 

x—2.     This  gives  ^       ^J\    ^    J 

4(2) 
To  find  B,  substitute  x  =  —  2,  omitting  the  factor  x-\-2. 

^^4-12-8^      o 

-4(-2)  -• 

To  find  C,  substitute  x  =  0,  omitting  the  factor  x. 

—  S 

—  4 


EXAMPLES 

The  constant  of  integration  C  will  be  omitted  in  the  examples  in 
this  chapter,  and  the  following  chapters  on  the  integration  of  func- 
tions. 


J      r      X*  dx       _  ^3      3^ 

■  J  x--'6x  +  2~  '^        2 


+  1  X  +  l0£ 


.c-1 


{x^-l)(x-^f 

r 


\J    2      f  (•^•-+  X  +  l)dx   ^  1  ^     (x  +  1)  (x 
•  Jar'- 4  0.-^  + a; +  6      12     "         (.«-2 

\     3_     r0v+iydw^l^^J2w  +  l)(2w-l) 
'  J     4  z(j^  —  w        8    "^  w^ 


INTEGRATION    OF   RATIONAL    FRACTIONS  253 


"4.  r ^ ^^loc,   (^+^y  . 

J  (.1-  +  1) ( .r  +  .'Jj  (.f  +  5 j      8    "^  (.1-  +  1  )(.f  +  o)* 


J  (2  .r  - 1)  (3 .0  - 1) (3  X  - 2)      18    ^         (2  x  -  If 


g      r ax  +  hx ,1^.^^ — loa  (6.^;  _  a)+ — ^1 — log  (ax-b). 

J  {ax  -  b){bx  -  a)         ab  -  b-    '^  ^  ^     ab  -  «-    °  ^  ^ 


r     (x  +  ^ifdx  ^    ^  1  ^^J2x  +  a){x-ay^ 
J  2  x^—  a.i^  —  d-x      G     '^  x^ 


/(x±a±Jjfdx 
{x  +  a){x  +  b)  ' 


X  + log  (x  +  a)  +  -^  log  {x  +  b). 

b  —  a  a  —  b 


10 


r(a^--b')(x'  +  ^)fb-  ^     1    j^^,/^/x  +  ?>)a>.r-f>)_ 
J  (a'X^ —  b-){b-x- —  a'-)      2  (ib     "^  (^ax  —  b){bx +  a) 


11    r       (x  +  i)<Jx  1  ^^^(x  +  r^nx-iy 

'  J  (x^-  19)--  4  {x  +  8)-     ;iOO     "'  (x  +  3)''(x  -  1)'  * 


j2     i"       (.T  +  l)dx 
■  J4ar^-17ar'  +  4.c 

=  T^^*^^^^^'-^^'^S[(2x  +  l)(2.-l/]  +  llog, 
1 20         .r  +  2        lo  4 


254  INTEGRAL   CALCULUS 

198.   Case  IT.    Factors  of  the  denominator  all  of  the  first  degree,  and 
some  repeated. 

Here  tlie  method  of  decomposition  of  Case  I.  requires  modifica- 
tion.    Suppose,  for  example,  we  have 


If  we  follow  the  method  of  the  preceding  case,  we  should  write 
x^  +  1        A  ,      B      ,      C  D 


x(x  —  If      X       x  —  1      x~l      x  —  1 

But  since  the  common  denominator  of  the  fractions  in  the  second 
member  of  this  equation  is  x{x—  1),  their  sum  cannot  be  equal  to  the 
given  fractisn  Avith  the  denominator  x{x  —  iy.  To  meet  this  objec- 
tion, we  assume 

ar'  +  l         A  ,        B       ^       C        ,      D 


x{x-lf      X      {-c-lf      {x-iy-      x  —  1 

Clearing  of  fractions, 
ar'  +  1  =  A(x  -  If  +  Bx  +  Cx(x  -  1)  +  Dx{x  -  If 

=  (J  +  D)x'  +  (-  3  .1  +  C -  2  D)x'  +(^SA-\-B-C+  D)x  -  A. 

Hence  A  +  D  =  l, (1) 

-3^+(7-2Z)  =  0, (2) 

SA  +  B~C+D  =  0, 

-.1  =  1. 

Whence  A^ -\,  B  =  2,  C  =1,  D  =  2. 

Therefore        -^^^,  =  -  -  +  -^  +  r^'  +  ^- 
x{x  —  ly  X      (x  —  l)"    (.i;  —  1)-      x  —  1 


! 

IXTKfiKA  riOX    OF    HAIIOXAI.    FILVCIIONS  255 


The  numeratovs  ^1  aiul  B  may  be  detevuiined  by  the  short  method 
given  for  Case  I.,  and  then  C  and  D  uiay  be  found  by  (1)  and  (2). 


EXAMPLES 

rx'-x'  +  l  ,        .r\  1,1,,      x-1 

J       x*  —  x'^  1  Ix-      X  X 

o      r         x-dx  2-3  X     ,  1  ,      .r  - 1 

^  J  {x  +  l){x-lf     4(.f-l)-      8     ""x+l 

„      f     8  cJx      _  1        ,  1  ,    ,     .r- 

^    "^^   Jxix^-^f-~¥^A:^l^'''^'^f^^' 

4      f     (10  g.-- 32V?..     ^         1  ,  3i^,2x-3. 

■   J  (4ar-|-l)(2a;-3)-      4(2a;-3}      4     °4x  +  l 

J  (9  .«-'  -  4;-  ~      1 8(9  x"  -  4)      2 10   *^^'  3  x  +  2 " 


^  (a;-  —  t(.r)-  jr  —  ax  x  —  a 

7.     ff'^  +-;Y';..-=  .X  +  '4!^|)  -  ^  --1.  +  8  log  (.  -  ,). 

./  \x—  \  :\f.r  -A  )'        .'•  — 1 


INTEGRAL   CALCULUS 


X—  2  —  Vo 

J  {x+a)Xx  +  by  {a~by\x-\-a      x  +  bj 

H lot?  (x  +  a)  H loff  (a;  +  6). 

199.   Case   III.     Denominator   containing    Factors    of    the   Second 
Degree,  but  none  repeated. 

The   form    of    decomposition    will    appear    from    the    following 
example, 

Jx(x'  +  'i)  '^' 

^TT                            OX  +  12      A  ,  Bx  -\-C  ..  s 

We  assume  ^  ,        -  =  —  H , (1) 

a-(.r  +  4)       X        ar-^i 

and  in  general  for  every  partial  fraction  in  this  case,  whose  denomi- 
nator is  of  the  second  degree,  we  must  assume  a  numerator  of  the 
form  Bx  +  C. 

Clearing  (1)  of  fractions, 

5  X  +  12  =  (.4  -[-  B)x'-  +Cx  +  4A. 

A  +  B^O,  C=r>,    4.4  =  12. 

Whence      A  =  3,  B  =  -3,  C=5 ; 

r>x  +  12      3  ,   -3.T  +  5 


therefore 


x{x;-  +4)      x         XT  +  4 

J      X-  +  A  J  x'  +  4:       J  a^  +  4 

=  -|log(.T^  +  4)  +  ^^tan-|. 


INTEGRATION   OF    RATIONAL    FRACTIONS 
J  x(x-  +4)  °  Var'  +  4      2  2 

Take  for  another  example, 

r   (2  x""  -  3  X  -  ^)dx 
J  {x-l){x--2x  +  5)' 


This  fraction  is  decomposed  as  follows 


2x-~3x-P>  1       ,       3.r-2 


T  1  1—   ' 


{x  —  l){x^  —  2x  +  o)  x  —  1      ar  — 2a;  +  o 

/(3x-2)(lx  ^  r(3x-3)(]x        r         (Ix 
x^  —  2x-i'5~Jx-  —  2x  +  d     J  .r  — 2.f  +  c 


=  I  log  (x'  _  2  .r  +  5)  +  ^  tan->:^  ,^-i 


J  (x  -  1) (x-2  -  2  X-  +5)  *'         .r  - 1  2  2 

The  integration  of  any  fraction  with  a  quadratic  denominator  like 

the  preceding,      (   v  ^~  -^r  •*'  ^  uij^y  ]je  shown  as  follows: 
J  af  —  2x  +  5 

Having  written  the  denominator  in  the  form  (x  +  a)-  +  Ir,  we  have 

f  (px  +  q)dx  ^   rp(a;  +  a)(Za;        j"  (f/-pa)dx 
J  (x  +  a)-  +  b-     J  {x  +  a)-  +  b-     J  (.f  +  af  +  b- 


258  INTEGRAL   CALCULUS 


EXAM  PLES 


1.     I  ' —  ax  =  —  —  bx  +  ~  log h  3  Vo  tan^ 


V3 


4.     I =  — h-log h- tannic. 

Jx'-l      3      4     ='a-  +  l      2 


When  the  given  fraction  and  the  denominators  of  the  partial 
fractions  contain  only  even  x^owers  of  x,  they  may  be  regarded  as 
functions  of  xr,  and  we  may  assume  A,  B,  C,  etc.,  as  the  numerators 
of  the  partial  fractions. 

In  the  following  example,  the  partial  fractions  may  be  assumed  as 

^     +      B 


X-  +  a^     X"  +  &^ 


5.     f ^^ =  x  +  — 1— ^ftHan-i--anan-'^ 


.      _,       3x 
tan 


6-     f..    ^}-:r        =Ytan-^2.-tan-'g)  =  l„,__     ^      _ 
J  (4  x'  + 1)(.«^  +  4)      6  V  2y      6  2  +  2  .^•- 

J  {a-j?  +  6-)(6-'a;-'  +  ci-)  ah  ab(l  -  a;^) 


8.     C-i^-IlD^^  =Llogt 
J  x{a^-6x+l'3)      26     ° 


6.'K+13  ,    5  ,     _iX-3 


INTEGRATION    OF    RATIONAL    FRACTIONS  25'J 

9      r     (3.r^-2.T-20y.r    ^1  ^^^(2  x^ -6  x +  5^ 

J  {x- +  3){-j  X- -  (j  X  +  r))    4    '^      (.f-  +  a)- 

+  A  tan-i-^  -  ^  tail-'  (2  x  -  3). 
V3  V3     ^' 

J  l/'~l      6        y- +,'/  +  !      V3  V3 

11.  r(£MlgV^  ^  1 1,.  ^-^  +  -^  +  1  ^  Vgtan-^  i^^ . 
J  .*;■'  +  ■»-•'  + 1      4     ''x'-x  +  l        2  1  -  ar 

12.  r^^  =  ^-  log  "^-"-^g+^  +  -L^tan-  -^. 


13.     I = ■ — -+-log tan  ' 

J  ^^-x'  +  x^-l      G(.v  +  1)      4     °x+l      3v'3  V3 

200.  Case  IY.  Denominator  containing  Factors  of  the  Second  De- 
gree, some  of  which  are  repeated. 

This  cas*'  is  related  to  Case  III.,  as  Case  II.  to  Case  1.,  and  requires 
a  similar  modiiication  of  the  partial  fractions. 

For  illustration  take 

•^-'^  +  -  +  ^?x. 


-  r2_afj-ar+j 

J    (x^+iy 


We  assuir.a 


2aT'^  +  a;2  +  3  ^  Ax  +  B      Cx  +  D 

(ar'  +  l/         (p^  +  lf       ^  +  1   ' 

2  x'  +  .r  +  3  =  Cx'  +  ar"  +  (A  +  C)x  +  B  +  D. 
A=-2,  B  =  2,  C=2,  D  =  \. 


260  INTEGRAL   CALCULUS 


Therefore        ^l^f±^±l  =  ^ll±^  +  ^J^±l 
(x-  +  1)-  (.«-'  +  1)-        of  +  1 

r-2x±2^^^^_  r  2xdx       ^r     dx 

J  (x' +1)'  '   .  J  {x' + 1)'     J  {^ + ly 


^1        I   2  f      '^'^ 
x'  +  l  J   (:r  + 


If 


To  integrate  the  last  fraction,  we  use  tlie  following  formula  of 
reduction, 

C      dx      ^  1  r         X  .^  ^^  _  g.   r        dx       1  ^ 

J  (■x' + «-)"      2  (>i  -  1)  <r  [_{x^  +  «'-')"-^  "^  ^"    '        U  (x--'  +  a-)"-'J* 

/dx 
— .    '   ,      by  making  it 
(x-+aY     -^  * 

/dx  V      I      / 

— - — .    By  successive   applications   the   given 

integral    is   made   to   depend   ultimately  upon    | 


dx  1  •  1,    • 

which   IS 


-tan 
a  a 


This  formula  may  be  derived  as  follows  : 

r 1  =  A[a;(x-  +  rt-2)-«]  =  (a:2  +a2)-»  -  2  nx'^(x^  +  a2)-n-i 


dx 

=  (a;--^  +  «-)  -»  -  2  «  [(x-  +  a-)  -  a^]  (a;2  +  a^)-"-! 
=  (1  -  2  >j)(-«-  +  «-)~"  +  2  ncfi  (x-  +  a2) -»--!. 


Integrating  both  members  after  multiplying  by  dx, 

=  (1  -  2  n)  r ^—  +  2  na-  f — , 

2„«3r ^^ = ^] +  (2n-l)f— ^^-. 

Substituting  for  «,  «  —  1,  we  have 

2  (H  -  1)  rt2  r '^^1 = ? ■+  (2  n  -  3)  f- — ^^^ 


INTEGRATION   OF   RATIONAL   FRACTIONS  ^tjl 

Substituting  in  the  formula  n  =2  and  a-  =  1,  we  have  ^^ 

{x'  +  iy        x'  +  i^x-  +  i^ 

A  partial  fraction    of   the   form  ^^'  "*"  ^^ ,  hv  substitutinL' 

[(x  +  ay  +  fr]"'    - 

X  +  a  =  z,  becomes  -^^  "^  ^  ,^  ^^,  the  integration  of  which  lias  already 

been  explained.  v^  +    J 

For  example,  if  x  —  3  =  ^,      I  — ^ — ^       \^  ,  =  I  r-r-^ — , ^^^ 

4(2=^+3)-  J(;2-  +  3/ 

By  the  formula  of  reduction, 

r    dz     ^ir      z      .gf    ^^    1 

J  (^2  +  3)«      12L(2^  +  3)^  ^  J  (2^  +  3)-^J 

^    z      1  ir  ^     r^'^-1 

12(22  +  3)-      4    6  |_;2- 4- 3      J  2- +  3j 

^  +-l-tan-  ' 


12(2^  +  3)^  '  24(^-^  +  3;      24  V3  V3 

\-3f  12{z'-\-3r^S{z'  +  3)     3V3  V3 


Hence        r_(^:!L±lML_  = l^'>^--^>-^ 

^       J(x2-6x  +  12/      12  (a^- 6  a; +  12)2 


2(.-3)       ^_l_tan-»x:z3. 
^3(.'c2_6x  +  12)^3V3  V3 


INTEGRAL   CALCULUS 


EXAMPLES 


J  \x'-  +  ay  X-  +  a-  a 

/'4a,_f.3  4aT''  +  na;-2,       1      ,     _i2x 

3.     I  -^ — „dx— — —^ -H tan^ 

J  (4  x^  +  3f  8  (4  x^  +  Sy       16  V3  V3 

For  the  following  example,  see  note  preceding  Ex.  5,  Case  III. 

4       r36  x'  (x'  +  iy  +  25  .gr  ^^^^  ^  g--'  +  X 

J    (4a;-  +  9)-(9ic2  +  4)2   "  2(4.^•2  +  9)(9ar' +  4) 

,     1    ^       1     13a;      ■ 

5.     C^  +  ^^-'^Ux  = ^(-±J1^ 

J  (a^  +  4x  +  9y  2(a;-  +  4:r  +  9) 

+  i  log  (;«-  -f  4  o;  +  9)  -  ^^  tan-^  ^J^ . 


% 


6      f  (.r^-2)f?a;  _  x  +  A 

J  {x:'  +  x  +  l){^x^-\-x  +  2y-~      7(a;'  +  a;  +  2) 


+  ^tan-2^1-^tan-2^+l. 

7V7  V7         V3  V3 


^      r9x^dx    ^__Sx_^l^        (x+iy   +V3t,,-.2^r-l 

g      /• dx _        12  xr  +  36  ai  +  29 

•   J  [(.i,  4_  2)^  -  (.i;  +  ]  )^].^  ~      2  (2  X  +  3)  (2  X-  +  6x  +  5) 

-3  tan-' (2  a; +  3). 


CHAPTER   XXIII 
INTEGRATION   OF   IRRATIONAL   FUNCTIONS 

201.  We  have  shown  in  the  preceding  chapter  that  the  integral 
of  any  rational  function  can  be  expressed  in  terms  of  algebraic,  loga- 
rithmic, and  inverse-trigonometric  functions. 

We  shall  now  consider  the  integration  of  irrational  functions. 

202.  Integration  by  Rationalization.  Some  integrals  involving 
radicals  may  be  integrated,  by  reducing  them  to  rational  integrals 
by  a  change  of  variable.  This  is  possible,  however,  in  only  a  very 
limited  number  of  cases.     This  process  is  sometimes  called  intefjnir 

Jion  by  rationaUzation. 

p 

203.  Integrals   containing   (ax  +  by.     Such   an   integral    may   be 

rationalized  by  the  substitution   ax  +  b  —  z''. 


/X'dv 
' — ~ T" 
{2x  +  'Sy 


Assume  2x  +  3  =  z'',     x  =  ^^,     dx  =  ^^^- 


Then 


)    2      r>  r 


+^'Y~(^-^-+'^y^p^+^)K^^'-'^^-'+^n 


Sfz'_3z^ 
'SVT       2 


INTEGRAL   CALCULUS 


'Another  example  is  (  /» 

J  Va^  +  l/ 

Assume  x  =  z^,     dx~2z  dz. 


Then  r_Z^^  f2^^2  f  f  .^  -  .  + 1  _  J_  >. 

=  2[t-t^z  -log  (z  +  1)~\  =  x^  -  x  +  2  x^-  -  log  (a;^  + 1)"'- 

p  r 

204.  Integrals  containing  (ax  +  by,  (ax  +  b)',---.  In  this  case 
the  integral  is -rationalized  by  the  substitution  aa;  +  6  =  2:",  where 
11  is  the  least  common  multiple  of  q,  s,  •••,  the  denominators  of  the 
fractional  exponents. 

Take,  for  example,         f ^ . 

*^  (x  -  2)^  +  (a;  -  2)* 

Assume  x  —  2  =  z^,     dx  =  6  z^  dz, 

{x-2f  =  z%     (x-2)^  =  z\ 

^^  (x-2)^  +  {x-2f      J^'  +  ^*        -^2  +  1        J{  z  +  lj 

=  eft  -z  +  \og(z  +  1)1=  3(x  -  2)^-6(x  -  2)«  +  6  log  [(.t-2)*  + 1]. 


EXAMPLES 


1-     f    ^t da;  =  2Va;-2  + V2tan-\/ 


•T  V  35  —  2 


o      r  2  VI Q  7        (4:X-3yn4x-Sy  ,  4a;-3  ,    9"| 

3-    /j^p^^=  =  |(3x-2)?-(3.;-2)^+log(l  +  ^3^=2). 


\y 


INTEGRATION   OF   IRRATIONAL   FUNCTIONS  265 

■   J  (iy  +  iy~       24(42/  +  !)^ 
6      f 7^^ =  -^^ 4-21og(V2^;rri  +  l). 

^^l  +  2f^  2         J        4 

8.  Cx'Va^b  dx  =  ~(^-^+  ^)^  (15  alx-^  _  12  abx  +  8  ft^). 
J  105  a^ 

9.  f '^ ^2(3x  +  l)^-4tan-i^:^. 

rv.r  +  1-1  ^^^  _  2  V^n  -  log  (x  +  3)  -  2  V2  tan-\/^. 


10. 


11.     f         (^-^)^ =  3(2a.-3)U^log^^i^^i+^ 

*^  (2x--3)^  +  6a;-9     ^  ^  (2x--3)3' 


3V3^     /2.T-3)^. 
4  V3 


12.     f  ^^^        — 

-^  V2a;  +  1+Va;-1 


=  2V2  x  +  1-  2 V^r=n  -f-  V3(  COS-'  - — ^  -  cos 


.i4  —  x  1 1  —  a;> 


a-  +  : 


a- +  2^ 


266  INTEGRAL   CALCULUS 


^  (2x* 


clx 


2a;^+llog(2a;'-l)+^log(x^  +  2) 


■l){x^+2)  9  9 


16  V2^       1  a;* 

— ^—  tan- '— =. 
9  V2 


14.     r   fa  +  2)rfa   ^2Va;  +  l  +  ^log(-x  +  2-V.i-  +  l) 
•^  (x  +  lj'^  +  l  3 

o 

205.  Roots  of  Polynomials  of  Higher  Degrees.  —  In  the  rationaliz;\- 
tion  of  irrational  integrals  we  now  pass  from  roots  of  binomials  of 
the  first  degree  to  roots  of  polynomials  of  higher  degrees. 

Here  rationalization  is  limited  to  the  square  root  of  an  expression 
of  the  second  degree. 


206.    Integrals  containing  V;<;"  +  ax  +  b.      This   may  be    rational- 
ized by  the  substitution 


■\/x-  -f-  ax  -\-b  =  z~  X. 

For  example,  consider      i  —  • 

/  '^  a;  v'x"'  —  x-\-2 

If,  following  the  method  of  the  preceding  articles,  we  assume 


V  ar  —  x  -\-  2  =  z,     x^  —  x-\-2  =  z'^, 

the  expression  for  x,  and  consequently  that  for  dx,  in  terms  of  z, 
will  involve  radicals.     This  difficulty  is  avoided  by  assuming 


V aj^  —  X  -{-  2  —  z  —  X,     —x-\-2  =  z-~2 zx, 

cancelling  x'  in  both  members. 
z'-2 


■y/xr  —  x-{-2  =  z  —  x 


(Iv 

2(z'- 

z  +  2)dz 

{2z 

-ly 

z-  - 

-z  +  2 

INTEGRATION   OF   IRRATIONAL  FUNCTIONS  267 

Hence, 

C2z-iy    '  _r'2<h_ i_,  ^2-V2 

22-1     2z-l 


Substituting  z  =  Vx-  —  x  +  2  +.t, 


— _: =  —  log  ^- — -^-^ 

xy/x^  —  x  +  2      V^        Var^  —  a;  +  2  + 


2  +  a;-V2 
X+V2 


207.  Integrals  containing  V—ar  +  ax  +  b.     This  may  be  rational- 
ized by  the  substitution 

V—  X-  +  ax  +  b  =  V(«  —  x) (jS  -f-  X)  =  («  —  .t)2;  or  =((3  +  x)z, 

where  «  —  x  and  /?  +  x  are  the  factors  of  —  x-  +  rf.r  +  b. 

These  factors  will  be  real,  unless  ^/  —  xr  +  ax-\-b  is  imaginary  for 
all  values  of  x. 


Take,  for  example,  |  —^  • 


Assume       V2  +  x  —  x-=  V  (2  -  a:)  (1  +  a;)  =  (2  —  x)z. 

^     ,  /o  X    o^^  2  2^  —  1         ,  fiZfZz 

l  +  x  =  i2-x)z;    cc=-^^,    '''''  =  (^TTf' 


3z 


^2  +  x-  x'  =  {2  -  x)z  =  -r^' 
Therefore, 

f  ^^^ =  fJi^  =  J_logi^^2^. 

-^  a;V2-fa;-ar^     J22;--l      V2       2V2  +  1 

9^* 


r  da;  ^  J_ ^p„  V2  +  2a;-V2-a;. 

J  a;V2  +  a;-a2      ^2     "  V2  +  2 a;  +  V2^^ 


268  INTEGRAL   CALCULUS 


EXAMPLES 


/: 


dx  .      _iX  +  ^'' x~ -\- 4:  X  —  4: 


xy/x-  +  4:X  —  4: 


rVa^  +  4a;^^^ ^==  +  log{x^2+V^^T^). 

^  x-\-  \xr  -\-4lX 


s- 


dx 


(2  ax  -  x')^      «' V2  ax-x' 
*^  3a;  +  4a 


—  2  tan 

-x/i- 

X 

4 

V3 

tan~ 

■V 

3-3x 

V3  + 

X 

3  +  a; 

=  cos~^ 

2 

0 

V3 

COS" 

1   2 
3- 

a? 

-X 

-4»io. 

-+^ 

lOR 

3:2- 

_a 

where  z  =  x  +  Vor  +  a^, 


208.  Integrable  Cases.  —  The  preceding  articles  include  those  ] 
forms  of  irrational  integrals  that  can  be  rationalized.  In  general,  ; 
integrals  containing  fractional  powers  of  polynomials  above  the  first  | 
degree  —  except  the  square  root  of  polynomials  of  the  second  degree  \ 
—  cannot  be  rationalized,  and  cannot  be  integrated  in  terms  of  the 
elementary  functions,  that  is,  cannot  be  expressed  in  terms  of  alge-  2 
braicj  exponential,  logarithmic,  trigonometric,  or  anti-trigonometric 
functions. 


INTEGRATION   OF   IRRATIONAL   FUNCTIONS  209 

Every  integral  may  be  regarded  as  defining  a  certain  function. 
It  has  been  shown  in  Art.  192  that  if  f{x)  is  any  continuous  func- 
tion of  X,   I  f{x)dx  is  a  function  of  x,  which  may  be  geometrically 

represented  by  an  area   bounded  by  the  curve  y  =f{x) ;  but  this 
cannot  always  be  expressed  in  terms  of  the  elementary  functions. 


1 

CHAPTER   XXIV 


V  \ 


*  TRIGONOMETRIC  FORMS  READILY   INTEGRABLE 

209.     It  is  to  be  noticed  that  any  power  of  a  trigonometric  func- 
#    tion  may  be  integrated  by  Formula  1-.,  when   accomp9,nied  by   its 
dilferential 
Thus, 

^^^_cos"t^x- 


sm"  X  cos  X  ax  = ,       I  cos '  x  sin  x 

n+1         J  /i  +  ] 

/^     „          •>     7        tan"+^c       r     .„             ,     ,             cof'+^x 
tan"  X  see  x  fix  = ,     I  cot"  x  cosec  xax= . , 
n+1       J                                          w+1 

/,          ,        sec"+'a; 
sec"  x  sec  x  tan  x  ax  = , 
71+1   ' 

/„        *            .      ,            cosec"''"'a; 
cosec"  X  cosec  x  cot  xdx  = . 
n  +  1 

Having  in  mind  these  integrals,  the  student  should  readily  under- 
stand the  transformations  in  the  following  articles. 

210.    To  find  I  sin"a;dcf  or    |  cos"xdx.      When  n  is  an  odd  posi- 
tive integer,  we  may  integrate  as  in  the  following  examples: 


I  sin*xf?.r=  j  sin'*  x  sin  X  c?a;  =  j  (1  —  cos- a;)- sin  a; 
=  I  (1  —  2  cos-  x  +  cos*  x)  sin  x  dx 


dx 


,  2  cos''  x     cos^  X 

cosx'H ^ — 

3  5 


270 


TRIGONOMETRIC   FORMS   READILY    INTEGRARLE       271 

Another  exaini)Ie  is 

fcos"  2  X  dx  =  fcos^  2  x  cos  2  x  dx  =  j  '  C(f-  sin-  2  a;)  cos  2  a;  2  rfx 


=  ?/'sm2:.-«i^ 


211.   To  find    I  sin"*  a;  cos"  x  dx.     When  either  m  or  n  is  an  odd 

positive  integer,  this  form  may  be  integrated  in  the  same  manner  as 
in  the  preceding  article.     For  example, 

I  sin* X cos''  x  dx  =  i  sin*  x  co.s*  x  cos  x  dx  =  |  sin* x(l  —  sin^ x)-  cos xdx 

//  .  4  o  •  G  ,  •  ■!  \  7  sin^a;  2sin^a;  ,  sin^.r 
(sin*  .1-  —  2  sin" x  +  sm"  a;)oos  x  dx  = H 
^                                        ^                      5              7             9 

Another  example  is 
j  sin^  a;  cos  -  x  dx  =  j  cos^  x  sin'  a:  sin  x  dx  =  j  cos^  a;  (1  —  cos-  x)  sin  xdx 

//      4            ^   \  •       7           2  cos^  X  ,  2  cos^  X 
(cos^.T— cos^  a;)  sm  a;  dx  = ~ 1 
5               9 


*       r  •   7     1  ' ",        -i         3  cos'  .t'  ,  cos^ ; 

1.  I  sin^  a;  dx  =  —  cos  x  +  cos'  x ;; 1 ;;- 

J  i)  i 

n      r      <)     1         •  4sin''a;  .  Gsin^a;     4  siii^r  ,  sin^a; 

2.  I  cos-*  X  dx  =  sin  x 1 — . 1 ——  ■ 

J  3  5(9 

3.  fsin»  -  dx  =  -  2  cos  ^  +  ^  cos-"*  ^  -  p  cos^  f  • 

4.  J  s.n"^  COS'*  "*  =  -g^ jp'^  +  -^3-'  -  -TJ- 


272 

,5.     rsin^2^cos«2  6d^ 


intp:gral  calculus 

sm^2  0     sm^2  0 


12 


16 


6 .     I  (sin^  X  +  cos"  x)  sin^  x  cos-  x  dx 


cos^  X     4  COS"  X     6  cos^  a;      cos^  x 


7.     j  (cos''  <^  +  sin''  4,)  (cos^  ^  —  sin^  <^)  d<^ 


sin- <^  cos  ^  +cos-^sin<^  +  ^  (sin^<^  +  cos^  <;()). 
5 


g_     pin^  y  ^y  ^  cos^  _  Scos^^y  ^  3  cos^  i,  ^  ^^^  ^^^  ^ 
J     cosy  6  4  2 

q      /'cos^  X  d: 
J     sin*  a; 


dx       .        ,2 
sm  a;  + 


sin  x     3  §in^  a; 


10 


/cos''  cc  (7x  _      2  sin- 


a;  +  6 


Vsin^a;  SVsina; 


11 .     j  (sin™  x  cos^  X  —  cos™  x  sin^  ;*;)  rfa; 


sin'"+^  X  4-  cos'"^^  a;      sin^^"  x  +  cos"'^^  a;  ^ 
m  + 1  m  +  3 


12. 


13. 


I  (sin  2  a;  +  cos  2  x)  cos"  a;  dr  =  -  (sin-'  x  —  cos^  x)  +  sin  a;  cos^  x. 


Jsin 


4  a;  sin"  x  dx  ■■ 


4sin'''.T      8sin^.T 


TRIGONOMETRIC   FORMS   READILY   INTEGRABJ.E       273 
212.   To  find   i  tan"  x  dx,  or    i  cot"  x  dx. 

These  forms  can  be  readily  integrated  when  n  is  any  integer. 
I  tan"  X  dx  =  |  tan"~^  x  (sec-  x  —  T)dx 

=  I  tsiii"~-xseG-xdx—  I  tan"~-xda; 
—  -   rtan"-2a;< 


tan"- a;       ..-»-..^^. 


Thus   i  tan''iCf?.T  is  made  to  depend  upon   i  tan"  ^xdx,  and  ulti- 
mately, by  successive  reductions,  upon    |  tsmxdx  or   |  dx. 

For  example,      I  tan^  x  dx  =   j  tan'^  x  (sec-  x  —  1)  da; 

^tan^-_  r^auSicda;. 

I  tan^  x  dx  —  I  tan  a-  (sec-  x  —  l)dx 
=  ^^-\ogsecx. 

Hence  ftan'  x  dx  =  ^^'  -  ^^  +  log  sec  x. 

Another  example  is 

//*                                                      cot.^  X        {^ 
cot*  a;  dx-  =  |  cot*  x  (cosec-  x  —  V)dx  =  —  — p I  cot*  x  dx 

.=_22^_    rcot^^-(cosec^a.-l)d.r=-^:^'+^-^+  fcot^a-d.r 

_c_ot^_^o_ot^_^  f(cosec^a;-l)da.-=-2^^  +  ^-2^-cotx-x. 
5  3        »/  o  3 


274  INTEGRAL   CALCULUS 

213.    To  find    j  sec"  x  dx    or    |  cosec"  x  dx.     When  n  is   an  even 

positive  integer,  we  may  integrate  as  follows : 

I  QeG^xdx=  I  (tan^ic+l)-sec-a;dx=  |  (tan'*a;  +  2  tan^a;  + l)sec^a;da; 

tan^  X  ,  2  tan-''  x  ,  , 

=  — 1 1-  tan  x. 

o  o 


Another  example  is 

1  (cot'^x-A-T)  cosec^ xdx=  ~- 


I  coseC*  X  dx  —  I  (cot^  aj+1)  cosec^  xdx= ~ cot  x. 


214.    To  find  i  tan"*  x  sec"  xdx  or  |  cof"  x  cosec"  x  dx.     When  n  is 

an  even  positive  integer,  these  forms  may  be  integrated  in  the  same 
manner  as  in  the  preceding  article.     For  example, 

I  tan^  X  sec*  xdx=  |  tan*^  x  (tan-  x  + 1)  sec-  x  dx 

//.      a     ,   ,     R   \       ■,     ,       tan" a;  ,  tannic 
(tan**  X  +  tan"  x)  sec-  cc  dx  = 1 . 

When  m  is  an  odd  positive  integer,  we  may  integrate  as  follows : 
I  tan^a;  sec^  xdx=^  \  tan*  x  sec-  x  sec  x  tan  x  dx 

.       ,         =  j  (sec^  X  —  1)^  sec^  x  sec  x  tan  x  dx 

=  I  (sec''  x—2  sec*  x  +sec-  x)  sec  x  tan  x  d.r 


*W' 


sec''  X     2  sec^  x     sec^cc 


Another  example  is  1 

I  cof  X  cosec'^  X  dx  =  |  cot"  x  cosec*  x  cosec  x  cot  x  dx  j 


cosec^cc  ,  cosec^a; 


J/    s       4  N        4-  ^     cosec^a;  , 
(cosec''  x  —  cosec*  a;)  cosec  x  cot  x  dx  = ~ f- 


TRIGONOMETRIC   FORMS  READILY  INTEGRABLE      275 

EXAM  PLES 

(    1.     \\j'3.v^xax= — \-  —  tan  a;  +  x, 

%/  (  o  o 

/  2.  Jcot^  2  "^-^  =  -  ^  «ot'  I  + 1  cot^  I  -  cot^  I  -  log  sin^  |. 

3.     fsec^Oyc^y^^  +  ^^  +  ^^^  +  i^^^  +  tany. 
/  4.     I  cosec'*Sx(Z^  =  — -/ — ^; 1 — h  cot's x  + cot  3 a;  J. 

5.  I  (sec  X  —  tan  x)  sec''  ic  tan"*  a;  dx 

1  2  1 

=  -  (tan^  X  —  sec^a;)  +  ^  (tan''  a;+sec'  a;)  +  -  (tan^  x  —  sec'  x). 

6.  r(sec''<^+tan''<^)'' d<^ 

=  I  (tan^  <^  +  sec-'' <^)  +  I^H!^  -  2^?^  +  2  tan  <^  -  <^. 
5  o  3 

M      Aan''a:+1  ,         tan"  a;      tan*  a;  ,   ,  ,  , 

7.  I  ' —  dx  = h  tan  x  +  loer  cos  x. 

J  tan  a;  + 1  5  4  ^ 

STsec'  a;  -f  tan'  x  ,        ,     «        2  sec"*  a;  , 
I ■ dx  —  tan''  X f-  sec  x  +  x. 

J  sec  X  +  tan  x  3 

STsec"  a.'  4-  sec'  x  ,         tan^  a;         ^.o      ,  o  i      4. 
.     I ^ fZa;  =  — cot-  ;i-  +  3  log  tan  x. 

J        tan^  X  2 

10.     I  — ^- — — —  dB  = ,  (tan-  B  —  sin-  B)  -h  log  (sin  ^  tan  6). 

J    cosec-^cot*^  2^  J-r     o  K  J 


276  INTEGRAL   CALCULUS 


11.  Jvs 


2^x  tan  X  (V sec^  x—  Vtan^  x)  dx 


2  '  r  9  3  s 

=  -  (tan  2  cc— sec^  *  )  +  ^  (tan^  x  +  sec=*  x). 


12.     I  (sec™  a;  tan^  x  —  tan"*"'  x  sec^  x) dx 


_  sec'"+^  a-  —  tan'"+^  x  _  2  (sec'"+-  x  +  tan"'"^-  x)     sec"'  x  —  tan"*  x 
711  +  4  ??i  +  2  ??i 

A' term     |  sec""  ;r  cosec"  a;  fZ.r      may  be   integrated,  when   m  +  n  is 

even,  by  substituting    coseca;  = . 

tana; 

1  o       r      5  s      7        tan*  X  ,  3  tan'  a;      cot^ »  ,    o  i      j. 

13.  I  sec*  X  cosec-  x  dx  = 1 +  3  log  tan  x. 

J  4  2  2 

14.  I  (sec*  X  —  cosec-  a;)^  dx 

tan^  X  ,  3  tan^  x  ,  tan^  x      „  ,  cot^  x  ,       , 

= 1-  - — 1 3  tan  X h  cot  x. 

7  5  3  3 

215.  To  find  |  sin"  a;  cos"  a;  dr  by  Multiple  Angles.  The  integra- 
tion of  this  form,  when  either  m  or  n  is  odd,  has  been  given  in  Art. 
211.  The  following  method  is  applicable  when  m  and  n  are  any 
positive  integers. 

By  trigonometric  transformation  sin'"  a;  cos"  a;,  when  m  and  n  are 
positive  integers,  can  be  expressed  in  a  series  of  terms  of  the  first 
degree,  involving  sines  and  cosines  of  multiples  of  x. 

If  we  use  the  method  of  Art.  211  for  integrating  terms  with  one 
odd  exponent  occurring  during  the  process,  the  following  formulae 
for  the  double  angle  will  be  sufiicient  for  the  transformation  of  the 
terms  with  even  exponents : 


>        TRIGONOMETRIC   FORMS   READILY   INTEGRABLE       277 
^         ^  sin  X  cos  a;  =  -  sin  2  x, 


'•        ^7  sin^  a;  =  -  (1  —  COS  2  a:), 

cos-  X  =^(1  -\-  COS  2  x) 


.«' 


For  example,  required    |  sin*  .r  cos-  x 


dx. 


siu*a;  cos^  x  =  (sin  x  cos  .r)-  sin^  x  =  -  sin-  2  x(l  —  cos  2  a;) 


-  sin^  2  a*  cos  2  a;  +  —(1  —  cos  4  x). 
o  16 


TT  f  •   4         ■>     7  sin^  2  a;,    x       sin  4  a; 

Hence  I  sin*  x  cos-  a;  clx  = • 

J  48         16         64 


EXAMPLES 


1.    Jsin*x-cZ..  =  ^('?^^-sin2^-  +  ?^Y 
y    2.      rcos''a;da;  =  -('''^^^'  +  sin2 
3.      I  sin^  X  cos^  a;  dx 


•  ,  sin  4  X 

x  +  — - — 


1  /  ,  _  sin  4  X 


S\  4      , 

sin32a;  .  3 


A       C  •  6     1         1  /-         A    ■    o      ,  sin32a;  ,  3  .     , 

4.      I  snr  X  fZ.f  =  -—  (  o  x-  —  4  sin  2  x  H f-  t  sm  4 

J  16V  3  4 


278  DIFFERENTIAL   CALCULUS 

5.     fcos" xdx  =  ^f5x  +  4:  sin  2x- '^'"^^  ^  +  | sin 4 a; 


6.      I  sm^  X  cos^^  da;  =  — -  3  X  —  sm  4  «  H 

./  128  \^  o 


7.      I  cos®  iK  sin-  X  dx  = (5x  +  -sin^  2  a;  —  sin  4  cc 

»/  128  V  3 


8.      fsin*  xdx=^ (^ -  4  sin  2  a;  +  isin32  a;  +  ^  sin  4  x 
J  16  \^  8  o  8 


sin  8  a; 
"^     64 


CHAPTER   XXV 
INTEGRATION   BY   PARTS.     REDUCTION   FORMULA 
216.   Integration  by  Parts.     From  ilie  dil'ieiential  of  a  product 
d{\Lv)  =  M  dv  +  V  d7l, 

we  have  w:  =  I  " ''''  +   )  ^'du. 

Hence  I  ndv=^  ur  —  i  cdn (1) 

This  formula  expresses  a  method  of  integration,  wliich  is  called 
integration  by  parts. 
For  example,  let  us  apply  it  to 


1  X  log  X  dx. 

Let 

w  =  1  og  a-,     tla  en     d  v  =  x  dx ; 

whence 

,        dx              ,                a?- 
au  =  — ,        and        ^J  =  — • 

Substitutin 

g  in  (1),  we  have 

Jlogx-xdx  =  ]ogx-^-  j^y—- 

=  f.o..-J. 

(^) 


279 


280  INTEGRAL   CALCULUS 

Integration  by  parts  may  be  regarded  as  a  process,  which  begins\ 
by  integrating  as  if  a  certain  factor  were  constant. 

Thus  in  (2),  if  ixT  |  log  x-xdx  we  treat  log  x  as  if  it  were  a  con- 
stant factor,  we  obtain  log  .r  •  — •  From  tliis  we  must  subtract  a 
new  integral   formed  as  indicated  by  the  following  connecting  lines. 


I  log  X  •  X  dx  =  log  X-  —  —   j  — 


'x-  dx 

X 


This  method  of  remembering  the  process  may  be  found  useful. 
0 

Another  example  is  I  xcosxax. 

Assuming  ti  =  cos  x,  we  have 

i  xcosxdx  =  cosx  •  ~  —   I  ^(— since  dec). 

As  the  new  integral  contains  a  higher  power  of  x  than  the  original 
integral,  nothing  is  gained  by  this  application  of  the  process. 
But  if  we  take  ?t  =  x,  we  find 

I  X cos X dx  =  X sin x—  i  sin x dx 

=  X  sin  X  +  cos  x. 


EXAMPLES 


Cx  (e"^  +  e-"^)  dx  =  -  (e""  —  e""")  -  \  (e"'  +  e""^). 

I  X (sin 3 a;  —  cos 'Sx)dx=(  —'--}--\smox  —  (--}--] cos 3 ; 


INTEdRATIOX    HY   I'ARTS.      REUUCTIOX    FORMULA      281 

4.     fxf  logf-  +  smfjdx  =  ^  log  ^^  - 1'  -  2  j;  cos  I  +  4  siu  ^. 

6.  Clog  (ax  +  b)  dx  =  fx  +  -  J  log  (ax  +  b)-  x. 

7.  f(x  +  1)  log  (x  +  3)  dx  =  •'^•^  +  ^'  -  '^  log  (X  +  3)  -  ^  + 1 . 

8.  fsec*  (^  log  sin  </>  rf</>  =  /"^^yil^  +  tan  <l>\  log  sin  4,  -  ^^  -  ?^. 

9.  r^2i^cZ.  =  |V3^7^flog.-2)  +  i^tan-J 
-^  V3a;-2  3  V  J         S  "^ 

10_     rlog(..  +  2)^^^^.^_log(.r  +  2)      j^+1. 
J     (a; +1)2  a;  +  l  ^x  +  2 

11 .  I  tan~'  -  dx  =  a;  tan"'  - —  -  log  (.r^  +  a')* 
»/  a  a      2 

12.  fx-  tan-'  a;  dr  =  -  tan"'  a;  -  -  +  -  log  (a;-  +  1). 
J  3  GO 

13.  f  sin  -'  -  dx  =  x  sin"'  -^  +  Va'  -  ar\ 
J  a  a  .        . 

14.  r(3  a;2  -  1)  sin"'  a;  dx  =  (3^  -  x)  sin"'  x  -  ^^  ~  ^  • 


3a;-2 


282  INTEGRAL   CALCULUS 

,-       r      •   q     7           /cos^T              \   ,  sin^a;  ,  2  sin  a; 
15.     I  X  siir .T  ax  =  xi  — cos  x    -\ 1 — • 

X  (sec"  X  —  tan''  x)  dx  =  x  tan"  x  +  ~~  —^  +  log  sec  x. 
-  17.     fl2?i^l+i)  dx  =  x-'^^±^  log  {e^  +  1). 


18. 


j  log  (a  +  Viv^  +  «")  dx  =  X  log  (a  +  V.t^  +  a^) 


+  a  log  (.^•  -f-  V  ir  +  a^)  —  .r. 


In  each  of  the  following  examples  integration  by  parts  must  be 
applied  successively. 

19 .     Cx\-^-^  dx  =  -  —  (4  .r''  +  6  a^  +  fi .'«  +  3). 

(e2.  _  ,^yv7^,  =  ^  _  '1^(4  .^  _  1)  _|.  -ll.  (2 .t2  -  2  a;  +  1)  -  ^ . 

21 .  ra;"-^  (log  xf  dx  =  i^'  [(log  xf  -  ^-i°S^  +  -^1  • 
J  n  [_  It  irj 

22 .  fx''  sin  2xdx  =  f''^  -  ^^  sin  2 .«  -  /^^'-  — ")  cos  2  a;. 

23.  r.c(tan-^  xf  dx  =  ^—  (tan-^  x)-  -  a;  tan-'  x  +  ^  log  (.>r  +  1)  . 

24 .  Cx  ]  og  (.i-  +  a)  log  (a;  -  a)  dx  =  ^  ~  ^'  log  (.a)  +  a)  log  (.t  -  a) 

-(^^+^)-log(a.  +  a)-^-^^^'log(.T-a^+f- 
4  4  4  " 


INTEGRATION    BY   PARTS.     REDUCTION    FORMULAE      283 

217.   To  find         I  e'"^  sin  nx  dx,  and    |  c.'"  cos  nx  dx. 

Integrating  by  parts,  with  u  =  e"^, 

e'"  sin  ux  dx  — h  -  I  e"  cos  nxdx.     .     .     (1 ) 

Integrating  the  same,  with  u  —  sin?tx, 

e"'^su'ivxdx  =  - ■ I  e^'^cosnx  dx.     .     .     .     (j) 

a  aj 

We  see  that  (1)  and  (2)  are  two  equations  containing  the  two 
required  integrals,  j  e'"' sin  7(a;  (7.i-  and  |  (^'" cos  nxdx.  Eliminating 
the  latter,  by  multiplying  (1)  by  ir,  and  (2)  by  o^,  and  adding,  gives 

(cr  +  ?r)  I  e'"'  sin  ?ia;  dx  —  e"^  (a  sin  nx  —  n  cos  7fa;) ; 

,  f  ^^   ■  ,        ^''''(asin  ?/.r  —  ?icos7«a;)  ^ox 

hence  I  e""  sm  7(,<' f/.r  =  — ^^- ^ ^ -^ (o) 

^  a-  +  ?r 

Substituting  this  in  (1)  and  transposing,  gives 

(I  C  aj^  7        ('"■'((Uf  sin?(.T +  a-cos?j.'t') 

-  I  e'"'co?,nxdx  =  — ^^ -—^ ^; 

7iJ  (a-  +  ir)n 

,  C  „^  ,        e"^  (n  ?\\\  nx -^  a  0,09,  nx)  , .-. 

hence  I  e"^  cos  ?kt  a.t- =  — ^^ ^ '- (.•*) 

J  a-  -\-  II- 


EXAMPLES 


The  student  is  advised  to  apply  the  process  of  Art.  217  to  Exs. 
1-4.  For  the  remaining  examples  he  may  substitute  the  values  of 
a  and  n  in  (3)  and  (4). 


I     j  e""^  sin  5  x  dx  =  |-  (3  sin  5  a;  —  5  cos  5  x), 

/e^  cos  5  X  dx  =  ^  (5  sin  5  a-  +  3  cos  5  x). 


284  INTEGRAL   CALCULUS 


2.   \ 


e~^  sin  xdx= (2  sin  x  +  cos  a;), 

I     I  e~^  cos  xdx—  —  (sin  x  —  2  cos  a;). 

I  J  5  ^ 

Je'"'  sin  ax  dx  =  —-  (sin  ax  —  cos  ax). 
2  a 


i^     _?  /y»  V\  P     ~  f  X  X 

4.     I  e  2  cos  -dx  =  TT^     (  2  sin  -  —  3  cos 


K      /^sin  2  a;  +  cos  2  a;,    _  _  sin  2  a; 


13     V         3  ?>j 

+  5  cos  2  X 


13  e^ 


(e^""  +  sin  2  a;)  (e^  +  cos  x)dx  = 1-  —  (sin  aj  +  2  cos  x) 


,  e""  /  •    o         o        o    \      2  cos'  X 

A —  (sm  2  a;  —  2  cos  2  a;) . 

5^  ^  3 


7.  fe-^  cos^  3  a;  cia;  =  —  +  —  (3  sin  6  a;  +  cos  6  x\ 
J  4      40 

o      C  X   •    ct      •    o     1        ^''T-         ,  5  sin  5  a;  +  cos  5  x\ 

8.  I  e*  sm  2  a;  sm  3  a;  dx  =  —    sm  a;  +  cos  x -^ , 

/g2i 
a;e^'^  cos  xdx  =  —  [5  a;  (sin  a;  -f  2  cos  a;)  —  4  sin  a;  —  3  cos  x']. 


218.  Reduction  Formulae  for  Binomial  Algebraic  Integrals.     These 
are  formula  by  which  the  integral, 


I  x^  {a -{- hx^'Y  dx, 

5gral,  with  either  m  or  p 
ch  formulae,  as  follows: 


may  be  made  to  depend  upon  a  similar  integral,  with  either  m  or  p 
numerically  diminished.     There  are  four  such  formulae,  as  follows: 


INTEGRATION   BY   PARTS.     REDUCTION   FORMULAE      285 

Cx"'(a+bx")Pclx 

{n2)  +  m  +  l)b         {np-^m  +  l)bJ  ^    ^       ■>        '  V    ^ 

I  a;™  (a  +  fto;")*'  da; 

7ip  +  m  +  l        njy  +  m  +  lJ       ^  ^  >  v    y 

Cx'"(a  +  bx")Pdx 

^ x--^(a  +  bx''y^ _  (np^n  +  m  +  l)b  T  „,„ ^^^      ^  ^ 

(?7i  +  l)a  (m  +  l)a        J  ^  ^    ^ 

ra;"(a  +  ?>.c")Pf?.^- 

^  _a-'"+H«  +  ?>-'^")''^^_^^'/'  +  "  +  >» +i  f.v'"(a  +  ?..")''^V?.r.     .     (D) 
n{p-\'l)a  n\p +  l)a     J 

Formulae  (A)  and  (B)  are  used  when  the  exponent  to  be  reduced, 
m  or  2>,  is  positive,  (^1)  changing  in  into  m  —  n,  and  (73)  clianging  p 
intop  —  1. 

Formulae  (C)  and  (D)  are  used  when  the  exponent  to  be  reduced, 
m  or  p,  is  negative,  (C)  changing  m  into  vi  +  n,  and  (Z>)  changing  ;* 
into  j>  + 1. 

If,  in  the  application  of  one  of  these  fornnd;o  to  a  particular  case, 
any  denominator  becomes  zero,  the  formula  is  then  inai)i)]u'able. 
For  this  reason, 

Formulae  (^1)  and  (B)  fail,    when   v])  +  /m  4- 1  =0. 
Formula  (C)  fails,      ,  when  m  +1=0. 

Formula  (D)  fails,  when  p-\-l=0. 

In  these  exceptionable  cases  tlie  required  integral  can  bo  obtained 
without  the  use  of  reduction  formula;. 


286  inte(;ral  calculus 

219.  Derivation  of  Formula  (A).     Let  us  put  for  brevity 
X=  a  -\-  bx",     (IX  =  nbx"~^  dx. 

X^dX 


Then 


Cx'^X"  dx  =  C.i 


nb 

Integrating  by  parts  with  u  =  a;™~"+',  we  have 

I  x"'X^  dx  = ; — -^^-  I  x"'  "X''+^  dx. 

J  nb  {p  +  1)      nb  (p  +  1)J 


(1) 


Comparing  the  integrals  in  (1),  we  see  that  not  only  is  w  diminished 
by  n,  but  p  is  increased  by  1. 

In  order  that  p  may  remain  unchanged,  further  transformation  is 
necessary. 

By  substituting     X"^^  =  (a  +bx'')  X^, 
the  last  integral  may  be  separated  into  two. 

A.,„-H^P+i  ^i^  ^  ^^  Cx'^-^X^'  dx  +  b  Cx'^Xp  dx. 

Substituting  this  in  (1)  and  freeing  from  fractions, 
7ib(i)  +  1)  Cx"'XP  dx  =  if™ -+iX^'+i 

.    —(m  —  7i  +  T)[ai  x^'-'^Xp  dx  +  b  i  x'^X^  dx\ 

Transposing  the  last  integral  to  the  first  number, 

{np  +  m  +  l)b  Cx^'^X''  dx  =  .'c™-"+iX''+i  -  {m  -  n  + 1)  a  Cx'^-^'X^  dx,  (2) 

which  immediately  gives  {A). 


INTEGRATIOX    BY    PARTS.      IlKDUCTION    FORMULAE     287 

220.  Derivation    of    Formula    (B).      Integrating    by    parts    with 

u  =  X'',     we  have 

f -"X"  dx  =  X"  ~ — -  -  (  -  -  •    pX"-'inx-'  (Ix 
m+1     J m + 1 

='^'-^rx--A>-d«. .  .  .  .  (1) 

m  +  l       m  +  lJ  ^ 

Comparing  the  integrals,  we  see  that  not  only  is  ^)  decreased  by  1, 
but  that  m  is  increased  by  n. 

To  avoid  the  change  in  m,  substitute  in  the  last  integral  of  (1) 

Z^x"=  X—a. 

Also  freeing  from  fractions, 

(m  +  1)  Cx'^X"  dx  =  x'^+^X"  -  vpf  Cx'^X"  dx  —  a  Cx'^X''-^  dx\ 

Transposing  to  the  first  member  the  last  integral  but  one, 

(np  +  m +1)  Cx'^X"  dx=x"'+^X''-j-npaCx"'XP''^dx,    ...    (2) 
which  immediately  gives  (B). 

221.  Derivation  of  Formula  (C).  This  may  be  obtained  frona(2), 
Art.  219,  by  transposing  the  two  integrals,  and  replacing  throughout, 
VI  —  11  by  711.     This  gives 

(m  +  1)  fi  CorX"  dx  =  .r'""'X''+'  -  {up  +  m  +  n  +  1)  b  Cx"'+"X''  dx, 

from  which  we  obtain  (C). 

222.  Derivation  of  Formula  (D).  This  may  be  obtained  from  (2), 
Art.  220,  by  transposing  the  two  integrals,  and  replacing  p  —  1  by  y. 
This  gives 

n(p  +  l)a  C^rX"  dx  =  -  .r'"+'X"+>  +  (up  +  n  +  m  +  \)Cx'"X'-^'dx, 
from  which  we  obtain  (B). 


288  INTEGRAL   CALCULUS 

EXAMPLES 

•^  Va-  —  a;-  ^  /-a 

Here      f-^^^  rxV-^')~^f^^- 
•^  Vet"  — aj^     *^ 

Apply  (^1),  making 

m  =  2,   w  =  2,  p  =  —-,   a  =  a^,   6=— 1. 


=  -^(«^-^^)^  +  fsin-^- 


2.    fV  rr  +  a;2  da;  =  I  Va^  +  ^'  +  '^'  log(a;  +  Vo^T^). 

Apply  {B),  making 

m  —  0,    n  =  2,    P  =  7T)    a  =  a~,    &  =  1. 

Aa^  +  a;^)^da;  =  |(a^+xO^  +  f  f-^^i 


dx 


fi 


dx         _Va;^'-«-'^J_gg^_i^_ 


x^-y/x^  —  c?        2  a-V-        2  c^         a 


INTEGRATION   BY  PARTS.     REDl'CTIOX   FORMULAE     289 

Apply  (C),  making 

m=-3,    n  =  2,    />  =  -  J,    a  =  -  a'-,    b  =  l. 

J  2  a-  2  a- J  ' 

2  a-x^        2o?         a 

A      C       dx  1  1       ..a; 

^  x(x-  —  cir)^  a- Vx-  —  a-     «'  « 

Apply  (Z)),  making 

3 

m  =  —  1,    n  =  2,    ^9  =  —  ^,    a  =  —  a-,    6  =  1. 


da; 


1  1       _,x 

.sec    -. 


a2(-^  —  or) 

\l  5.     f  VcT^^'  dx^-  Va^  -  a:-  +  ^  sin"'  H: . 
J  2  2  a 

'  J  Var  —  a^     -^ 

7.     fCa^  -  x^^  dx  =  ^  (5  a^  -  2  ar^  Va^^"^:^  +  ^  "' «'"  "'  ^ 
^  «/  8 


sin"':^. 
8  a 


8.     Cix'-a')^ dx  =  f (2 x2 _ 5 „2) v^2ir^_j. :3a  j^^ ^^ _^  Vx» - a= 
•/  8  o 


0- 


290  INTEGRAL    CALCULUS 

9 


/.^•^ Va-  —  x^dx  =  '-(2  ar  —  cr)  Vcr  —  x-  +  ^  sin~^  - • 
8  8  a 

10.     fxWx'  +  a-  da;  =  f  (2  a;2  +  a^)  Vrc^  +  a^  -  ^  log  (x  +  V^Ta")- 
»/  o  8 


"•/;:i 


dx 


(a'~x-)i      arVu'-x^ 


12.    Derive  the  formula  of  reduction  used  in  Case  IV  of  Rational 
Fractions. 

r    dx     ^       1       r      X         .^  ^^  _g.  r     dx     1 

J  (a;^  +  a-)"      2  (li  -  1)  tr  [ (.^-  +  a-)"-i     ^"  ^ J  (.c^ + d'y-'j 

,  13.     f ^^1—^ : 


dx  Sx""-  ft-      _  _3_  gg^_j  ^  ^ 

t'icVa;'  -  a'      2  a'^  a* 


/•       dx  2  ar  —  1     /  o  ,  -, 

14      1 z=z.  = V  x-  + 1. 

J  xVo;-  +  1         3  a;-* 


15      r      '^''  ^-^       -     x  +  3a       -,     3,(2  ^ 

^^-     )—===— :z — V2ax  —  a;-  +  — —  vers  '-. 

•^   V2  ax  —  X-  -^  2  a 

Write        f      '^  ^^^       -  ^  r    •'^'^^     .  and  apply  (A)  twice. 
*^  V-^  c(x  —  x^      ^  V2  a  —  X 


■t  n      r dx _  V2  ax  —  a? . 

•^  x\/2ax-.x-2dx  ~  a^ 


17.  /V2- 


X  —  g 
2 


— ■_ —  V2  ax  —  x^  +  —  vers~^  ~  • 
2  2  a 


INTECUATION    BY    PARTS.      HEDIICTIOX    FOKMILK      201 

Write      J  V2  ax  -  oy^  dx  =j  Vu-  -  {x  -  af  clx,     and  substitute 
Ex.  5. 

18.     CxV2^^^^^dx  =  -^^'  +  "^-^'"Vtf^^^^^  +  ^^yevs'-r 

•>'  0  2  r( 


iQ       rV2  ax  — a."  dx        /k ^  ,  ,  .t 

19.     I  =  V2  ax  —  x^  +  a  vers~^  -  • 

J  X  a 

20      C     ^^  '^'^'       =  —  ^'""^  V^^  "-g  —  •*-"   t   (2  m  —  l)o  r     a;"-'  da; 
^  V2  ax-  —  ;ir  ^'i  wi        J  V2  a.c  -  x-^' 

21.   r -^^ 

^  x'"'V2  a.K—  X- 

_ _    V2  a.x-— .r'    ,       m  —  1       /» tf^ 

(2  7?i  -  l)a.^'"     (2  m  —  l)a  J  a;"*"  V2  a.r  —  a^' 

22.  I  x"'V2  ax  —xrdx 

x'"-'(2ax-x^f'  ,(2m  +  l)a  C  ^  .    /- -„  , 

= ^ — --^ — ^  +  —r-  I  x,""    V2  a.c  -  ^  dx. 

m  +  2  ?ji  +  2     »/ 

% 

23,  I  V2  gx'  — x'-(?a; 


^       (2aa;-ar^^  m-3      rV2ax--a;'fZa;. 

(2  m  —  S)oa;"'      (2  m  -  {^)aJ  a-"-' 

223    Trigonometric  Reduction  Formulae. —Tlie  methods  oxplninod 
in  Arts.  211,  214,  are  apiiliciibic  only  in  certain  cases. 
By  means  of  the  following  lormula-, 

^  I  sin"*  .r  cos"  a;  f?.r,  |  tan"  x- sec"  x  rf.i-,  and  |  cot"' .r  cosec"  a;  rfa;, 

may  be  obtained  for  all  integral  values  of  m  and  w,  by  successive 
reduction. 


292  INTEGRAL   CALCULUS  j 

fsin-  X  cos"  X  dx  ^  -  ^^^""^  ^  "^^"'  "^  +  '^^^-^  Csm-^xcos^xdx.  (1) 

J  m  +  ri  m  +  n«/ 

I  sin™  x  cos"  a;  da; 

i 

Wl  +  1  7)1  +  1      J 

I  sin™  a;  cos"  x  dx 

^  _  sin"'+^a;cos»+^^'     m  +  ^.  +  2  T  .^^  ^  ^^g„+2  ^  ^^_      _  ^4^ 
n  +  1  H+1     J  '^ 

fsin- :.  da;  =  -  ^"^""^  -^  ^"^  ^  +^^^^^  fsin-^  a.  dx (5) 

J  m  m   J 

K 

rcos»a-da;  =  ^^^^^*^^^  +  '^=^rcos»-a;da;' (6) 

J  n  n    J  j 

ftan™  a;  see"  x  dx  =  ^^^1^1^ ^^iZzJ-  ftan™-  a;  sec»  x  dx.  (7) 

J  711  +  7i  —  1         m  +  n  —  lJ 

I  cot"  a;  cosec"  a;  da;  '\ 

V 

=  _  cot"-^a;cosec":g m-l_  r^^^_, ^ ^^^^^„  ^^^_     _  (8) 

m  +  w  —  1  m  +  ?i  — 1«/  i- 

/,        sec""-  X  tan  a;  ,  ?j  —  2  /*  „„„_2  ^.  ^^  /QN 

sec" a; da;  = 1 I  sec"  "'a; da; (.»; 
n — 1             n— U 


I 


INTEGRATION   BY   rARTS.     Ki:i>i:CTION   FORMUL/K     298 
Jcosec":.d.  =  -22!!£j±^  +  2^Joosec.-'xdx.    .    .    .  (10) 

I  t2in"'xdx= — — -^ —  I  tau"~^ifda;. (11) 

/,„     ,            cot"-^^•       r     4-n  "     7 
cot" xclx  — I  cot"~-xdx (11' 1 

224.    Derivation  of  the   Preceding  Formulae. — To  derive  (1),    we 

integrate  by  parts  with  u  =  siu"*"^  .i-. 

fsin'"  X  cos"  xdx  =  -  sin'"-^:?;cos"+'.r  ^  in_-l  r.^„-2  ^  ^^^gn+o  ^  ^j^ 
J^  n  +  1  ?t  +  U 

I  sin'""-a;  cos""*"-*  f7.v=  \  sin'"~-a;  cos".'ccZ.c—  |  sin"';>;cos".rf?j;. 

Substituting  this  in  the  preceding  equation,  and  freeing  from 
fractions,   we  have 

(771 +  n)  I  sin"*  .r  cos"  a;  rtic 

=  — sm"'~'a;cos"+*a;4-(77t  — 1)  |  sin"*^^  a;  cos"  oj  dx, 

which  gives  (1). 

To  derive  (2),  integrate  by  parts  with  xl  =  cos""'  x,  and  proceed  as 
in  the  derivation  of  (1). 

Formula  (3)  may  be  derived  from  (1)  by  transposing  the  integrals, 
and  replacing  7?i  —  2  by  m. 

Formula  (4)  may  be  derived  from  (2)  by  transposing  the  integrals, 
and  replacing  77  —  2  by  n. 

To  derive  (5),  make  ii=()  in  (1);  and  to  derive  ((')),  make  7/<=0 
in  (2). 

The  derivation  of  (7),  (8),  (9),  and  (10)  is  left  to  the  student.  ^  We 
have  already  derived  (11)  and  (12)  in  Art.  212. 


294  INTEGRAL   CALCULUS 


EXAM  PLES 

-      C  ■   &     1           coscc/sin^-T  ,5.3      ,  5  .      \  ,  5x 
1.     I  s\\fxclx  = 9~(  ^^ hr7,siira;  +  -sma;j  +  — • 


3 


n      C         17  COS  a;/    1       ,        3     \   ,  3,      ,      x 

2.  \QO&ed'xdx  = —   -^7-+^   .   ,      + -log tan-- 

J  4    \siira;     2&in^xJ      8  2 

3.  Cse^^xdx=^^^(^^+-^+l\ 
J  2  cos^  ic  \3  cos*  a;      12  cos- a;      8/ 


4.     I  cos" xdx—  — 


5 

+  —log  (sec  X  +  tan  a?). 


cos'  a;  +  -  COS'''  x  +  -—  cos^  ^-\~zr^  cos  a; )  4- 


6  24  16         7      128 


c      r*  •  4         o     ,        cos  X  /sin''  a;      sin^  x  sm  a;\  ,  a; 

5.     I  sm^  X  cos-  X  dx  = f ■ ]  -\ 

J                                 2    \    'S           12  8    y  16 

c      /^cos^a;  ,        3cos;f  — 4  cos^a;      3  cos  a;  ,3,       .  x 

./  surx'                   4 sura;              bsm-a;  8  2 

7      C       '^^         ^1/1 
J  sin'*  a;  COS''' ;c  cos-aj\3sin'''a; 


,       5  5  . 

3  sm  X     2 


+  -  log  (sec  a;  +  tan  x). 


8.     ftan^a;  sec^  x  dx  =  (^^^^  -  ^^^  sec''  x  + 


sec  X  tan  a; 


H log  (sec  X  +  tan  x). 

16 


9.    fcot^ X  cosee'xdx  =  g^^  '^^"^"^  ^Y-  "^^"^'  -^  +  '"^"'' ''  +  - 
J  2V3128 


-ilogtan| 


CHAPTER   XXVI 
INTEGRATION   BY   SUBSTITUTION 

225.  The  substitution  of  a  new  variable  has  been  used  in  Chapter 
XXIII,  for  the  rationalization  of  certain  irrational  integrals.  "We 
shall  consider  in  this  chapter  some  other  cases  where,  by  a  change 
of  variable,  a  given  integral  may  be  made  to  depend  upon  a  new 
integral  of  simpler  form. 

We  shall  first  consider  some  substitutions  applicable  to  integrals 
of  algebraic  functions,  and  afterward  those  applicable  to  integrals 
of  trigonometric  functions- 

226.  Integrals  of  form  i  f(x-)x  dx,  containing  (a  +  bx^i.  One  of 
the  most  obvious  substitutions,  when  applicable,  is  xr  =  z. 

By  this,  any  integral  of  the  form     if{x^)xdx 

is  changed  into  -  |  f(z)  dz. 

p 
Integrals  containing  (a  +  b.xr)9  are  often  of  this  form. 

Take  for  example  I  —  • 

J  VI  -x' 

By  the  substitution  x^  =  z, 

/x^  dx     ^  1  r    z^z 

This  is  of  the  form  of  Art.  203,  and  is  rationalized  by  putting 
1  —  2  =  tv\ 


296  INTEGRAL   CALCULU8 

The  two  substitutions  in  succession  are  equivalent  to  the  single 
substitution  1  —  x-  =  iir. 

Applying  this  to  the  given  integral, 

ar  =  1  —  id^,     X  dx  =  —  w  dw. 
r^dx^^_   r(l-tv^wdw^_   nl-^c^-)dw 


=  -(„-|)=-L"(,3_^=)  =  _Vl_ri(^  +  2) 


EXAMPLES 


-^  V2a;'-'+l  30 

2.  Cx' (a-  -  x"-)idx  =  —(6  x"  -  a^v'  ^  5  a^a"  -  x^)K 
J  132 

o      r       dx     -         1   T       V^'^  +  cr  —  a       1   ,  x? 

3.  I  =:r=r=  = log  =  - —  log   — 3::=3;3 

•^  a;  Va;''^  +  a-'      2  a        V.^'^  +  d-  +  a      2  a        ( Vcc'  +  a-  +  a^ 


1 1  X 

log 


a        Va;2  +  a^  +  a 


4.  r--^^?^--=2r(:?^+(a.^+i^ 

^  V  a;-  +  1  —  l-^L-'  J 

5,     f ^=^  =  ^log(V3:^^  +  l)+^log(V3"=n^^-3). 


227.  Integration  of  Expressions  containing  Vcr  —  a;-  or  Va;-  ±  a-, 
by  a  Trigonometric  Substitution.  Frequently  the  shortest  method  of 
treating  such  integrals  is  to  change  the  variable  as  follows: 


INTEGRATION   BY   SLBSTITL'TION  297 

For   Va-  —  x^,  let  a-  =  a  sin  6   or  x  =  a  cos  $. 


For   Var  +  a-,  let  a;  =  a  tan  ^  or  x  =  o  cot  6. 
For   Var'  —  a-,  let  a;  =  a  sec  ^  or  x  =  a  cusec  ^. 
do; 


For  example,  find 


Let  a'  =  a  sin  ^,  dx  =  a  cos  ^  dO, 

a-  —  ar  =  a-  —  or  surO  =  cr  cos'^. 

/dx       ^  racosddO  ^  J^  r   c?^    _  tan  0  _         x  ^ 

(a--x^y     •^    a^'cos''^       oV  eos"'^  ~    cr    "aVo^^^u? 

• 

a"  Va^  +  a^ 

Let  a;  =  a  tan  6. 

/dx        _  r     a  seer  Ode     ^  1  Tsec^  ^j^  ^  1  r  dO 
x-y/y?  +  a^     *^  atanO  -asecd      a  J  ta,n6     •      a  J  ain  $ 


It      /  n  i.  A\      1 1       V a-  +  a-  —  a 

=  -  log  (cosec  e  —  cot  0)  =-  log =!^^ 

u  a  X 

Again,  find  '  i  ~  -  da;. 

Let  a;  =  a  sec  6. 

C^^?:^dx=  r'taud-asecet^nedd^^  ftan^^r/^ 
J         X  J  a  sec  6  J 

=  a  C(see^e-  l)de  =  a  (tan  6-6) 


=  Va:^  —  w'*'  —  «  sec  '  -  ■ 
a 


298  INTEGRAL   CALCULUS 


EXAMPLES 


Jv«= 


a-   ■     tX 


ar  ax  =  -  -ycf  —  a;-  -^ —  sm  ^  -  • 


,     r     cix 

^  x-^x'  +  a- 


V.B^  +  O? 


+ 

dx 


3.      C—^^—  =  log  (x  +  Va.-  -  a^). 
•^  V'.«"  —  a- 

^L   4       r        ^?-g        _      (2x-  +  a-)Va-—cif^ 
>.  5.      I  -^^ ^ —  dx  =  — - — ^ h  log  (x  +  yxr  +  a-). 


J        x*  3aV 

y       r       da;        ^  (2  a;'^  - 1)  V^^M^  . 

8.      r\/^^^(?a;=  f^^±l-  dx  =  V^^^  +  log  (x  +  V^<^^^^). 
*^    V  a;  -  1  »/   ^_^.2  _  ]^ 


9.     f ^i?^ =  V^- 

^  (a;  +  l)Va;--l       ^a;  +  l 

10.     f '^^ =  \/^^- 

n.      r ^':g ^  _  V3  +  2  a;  -  a;'-'  +  sin-^  ^^  ■ 

•^  V3  +  2x-a.-2  2 


INTEGRATION   BY   SUBSTITUTION  2'J9 


For  V3  +  2x-  X-  =  V4 -  (u;- 1)-,     let  a;  -  1  =  2  siii  6. 

22      r  ^^^  ^         a;  4-1 

J  (.x-2  +  2  a;  +  3)^      2Vdr^  +  2a;  +  3 ' 


For  Va^-f2x'  +  3  =  V(a;+l)'  +  2,      let  a;  +  1  =  V2 tan 


13.  r ^ 


(2  ax-  —  ar)^      «- v2  ax  —  x- 
^*-    /VfiH  '^-^  ^  V(J+^(ar+ 3)  +  2  log  (^/JT5+  V^i^+S). 


15 


/XT  dx  (3  a  +  a;)V2aa;  — a?-  ,  3a-  .  ,x  —  a 
— — — — — —  =  —  i ! — ^~ sm~' 
V2ax-;f^                         2                       2               a 


228.  Substitutions  for  the  Integration  of  Trigonometric  Functions. 
A  trigonometric  function  can  often  be  integrated  by  transfonniiig  it, 
by  a  change  of  variable,  into  an  algebraic  function.  For  this  purpose 
two  methods  of  substitution  may  be  used,  as  shown  in  the  two  follow- 
ing articles. 

229    Substitution,    sin  x  =  z,  cos  x  =  z,    or    tan  x  =  z. 

Consider,,  for  example,        C    sin  x  cos  x  dx 

J  1  —  sin  X  -f  cos-  X ' 

dz 
Let    sin  a;  =  z,    then   x  =  si  n~'  z,   dx  =  — z^z=r  • 

J    sin  a;  cos  .r  dx     _  r    zVl  —z- dz      __  r     zdz 
1—  sin  a;  +  cos- a;  ~~  J  1—2;  +  !  —z^  Vl^^     "^  2  —  2  —  2* 


r        zdz  ^2    r^/^  ,  1    r  dz 

J  (2  +  z)(l-z)          3J2  +  Z     3J1-; 


(2  +  z)(l-2)          3J2  + 

=  -|log(2  +  2)-hog(l-z)  =  -ilog[(2  +  siua;)=-'(l-sinx-)]. 
00  o 


300  INTEGRAL   CALCULUS 

EXAMPLES 

J  a-+b~ta,irx     d-  —  b-[_       a  \a  Jj 

-J  J  sin  4  ic  8        1  +  sm  a;     4  -y/2        1  —  V2  sin  a; 

Let    sin  x  =  z. 

4.     f-^ ^. =  liogEa^(l±^.  Let    cosx  =  .. 

J  sin  a;  +  sin  2x     3  (1+2  cos  xy 

*  \   K      /^  sin  a;  4-  cos  a;    ,        3  .r      1  ,       /  .        ,  o  \ 

Ni  5.     I  ■ dx  = log  (sm  x  +  2  cos  a;).  /' 

J  sin  X  +  2  cos  x  5       5 


Let    tan  x  =  z. 


/tana-     ,            ,     1    i      tana;  — 
_      f/x-  =  a;  H log 
tan  o  X                   Yo         tan  ;f  + 


1    ,      tan  a;  —  V3 


V3 


7.  Show,  by  transforming  into  algebraic  functions,  that  only  one  of 
the  following  integrals  can  be  expressed  in  terms  of  the  elementary 
functions.    (See  Art.  208.) 


/Vtan  X  clx  =  C^^^^^,  where    2;  =  tan  a; 

^  l+z- 

V  r  /-■ —  1    r  'Vzdz    r  zdz         ■. 

I  Vsma;da;=  I  —  =  I  —  ,  where    2  =  sin  a 

^  J  Vi-2^    J  ^v^ 


INTEGRATION   BY   yUBSTITUTION  301 

230.    The  Rational  Substitution,  tan  -  =  z.       By  this  substitution, 

2  ^  ' 

sin  a;,  cos  a;,  tanx,  and  dx  are  expressed  rationally  in  terms  of  z.     For 
2  tan  ^ 


2 

2z 

l+.ta.r| 

■1+^^- 

COS.'C  = 

1-tan'l 
l  +  tau=| 

tana; 

2ta„| 

2z 

l-tau'^^^ 

l~z' 

From  -  =  tan'  ^  z,    dx  =  -^^— „ 

2  '  1+z- 


It  follows  that  the  integral  of  any  trigonometric  function  of  x, 
not  containing  radicals,  may  be  made  to  depend  upon  the  integral 
of  a  rational  function  of  z,  and  can  therefore  be  expressed  in  terms 
of  elementary  functions  of  a;. 

231.   To   find     C- — — Applying   the   substitution   of   the 

J  a  +  6sina; 

preceding  article,  tan  ^  =  z, 

/,     2dz 
l+z-     _  r 2dz 
^  ,    2hz      J  a{l+z')  +  2bz 
1+2- 

/2  adz  _   r  2  adz 

a:'z^  +  2  abz  +  a:'     J  (az  +  bf  +  a' -  b^' 


302  INTEGRAL  CALCULUS 

If  a  >  6,  numerically, 


r- 

J  a 


tan-^    '^""^^    =  tan- 


+  6  sin  a;      Va^  — ft'"'  Va-  — 6^      Va-  — 6^  Va--6- 


If  a  <  &,  numerically, 


J  a-\-b  sin  a:     J   (az  + 


da;  /^  2  a  dz 1        ^  ^  az-\-h  —  V6-  —  o? 

hf  -  {b- - d')~  ^yrz^i    "=  az  +  h+^b^-d" 


a  tan  -  +  6  —  V&"  —  a^ 
1        ,               a 
-— -—  los 


^^'-«'        atan^  +  &  +  V6^^^ 


232.  To  find    f 

J  a-\-b  cos  X 


2dz 


r        c?a;  ^    I  1+z^  r  2dz 

Ja  +  6cosa;         f  b{l  —  z-)~  J  {a —  b)z^  +  a -\-b 

^     2      C_J^_ 
a  —  />  J    9  ,  «  +  6 


a  — 0 


If  a  >  6,  numerically, 


J  a  +  bcosx     a—b^a+b  y'^-j-^ 

=  — — :=::^:r  tan~'(  a/- — —  tan  - 


INTEGRATION   BY   SUBSTITUTION 

li  a<b,  numerically, 

dx ^ 2_  r       dz 


/dx        ^ 2_  r       dz 
a  +  b  cos  X          h  —  aj  ^1  _  b  +  a 


V6^ 


log 


b  —  a 

z-\lb  —  ff  —  V6  +  0 
z  V6  — a  +  V6  +  a 


303 


Vd^-a^ 


log- 


6  —  a  tan  ^  +  V &  +  a 


V6  —  a  tan  ;^  —  V6  +  a 


EXAMPLES 

Integrate  the  following  functions  by  means  of  the  rational  sub- 
stitution. 


/ 


dx 


5  —  3  cos  X     2 


-tan-^/^2tan^ 


h 


dx 


tan  X  4-  2  -  V3 


+  2  sin  2  a     2V3        tana;  +  2  +  V3 

)(  3.     f ^ 

(^         J  5  sin  tc  + 


3  tan  -  +  2 
12  cos  ic     13  ^°^ 


2tan5-3 


'J  a  sin  x  +  h  cos  ic      -» /«2  _l  7,2     ® 


6  tan  ■-  —  a  4-  \  'a'^  4-  i' 


bcosx      V^^l/     ^z,tan|-a-Va^6^ 


J  sin  ic  +  A 


lo" 


tan- 


1  +  tan* 


304  INTEGRAL   CALCULUS 

6.     f '^ =  tan-_J 

J  3  —  sm  a;  +  2  cos  x  2 


=  p  log 

sin  X  —  cos  a;      5  .      x  ,  ^ 

tan  -  +  2 


}.     I =  -  tan "-  —  —  log  I  1  +  tan  -  )  • 

J  (l  +  sina.-  +  cosa-)^      2        2      ^,^^^^x         ^\    ^        2) 


233.   Miscellaneous  Substitutions.    Varions  snbstitntions  applicable 
to  certain  cases  will  be  suggested  by  experience. 

The  reciprocal  substitution,  x  =  ~,  may  be  mentioned  as  simplify- 
z 

ing  many  integrals. 

EXAMPLES 
Apply  the  reciprocal  substitution  x  =  -  to  Exs.  1-6. 


3     r'V2a.T-a;-  ^^^  ^  _  (2  ctx  -  x^^  ^ 
J  x^  3  aaf' 


4.    I         ^a;        ^  1 J 


^■/ 


Off 


a; 


x  Va^  ±  a;2      a        a  +  Va^  ±  t^ 
ar)^ 


5_    r(a^'-.rV,^^^_3(a.>-a 
J         x'  8  a;^ 


INTEGRATION    BY  SUBSTITUTION 


305 


/. 


dx 


V8ar'  +  2a;-l 


3  a; 


„     r    a:^dx    _        8  6        ,      ^       .  i      /     , 


0- 

Let  x'  +  2  = 


^  ?t  +  J  ?i  +  1 

9       r       f^a;  —  ^  r      ^'^ 2&.r        ,   ,^ « +  I'y.f"] 

J.i'(a  +  &.r)^      a^  [_2(a  +  ?>a;)-      a  +  6.t'        "^      *'      J 


Let  X-  +  6  =  z. 


Let  a  +  bx  =  xz. 
10.     f — ?1I1^_  c^^.  =  sin  X  -  sin  a  log  tan'^^±^-  Let  x  +  a  =  z. 

^^-     I    ar  ,     2x  ,     ^=-.T-e  -  +  logVe--  +  e-  +  l -tan   ' — • 

Let  e"^  =  z. 


12. 


rJ^da;=V(a-rc)(6  +  .'«^)  +  (a  +  &)sin-»J^- 


Substitute  6  + a;  =  2-,  and   the   integral   takes   the   form   of   Ex.   5, 

p.  ^89. 


J3. 


^x  +  b 


dx 


=  V'(x  +  a)(:x  +  b)  +  (a-b)\og(Vx  -\-a  +  Vx  +  b). 


Substitute  x  +  b  =  z",  and   the    integral    takes   the   form    of   Ex.    2, 
p.  288. 


CHAPTER   XXVII 
INTEGRATION   AS    A   SUMMATION.     DEFINITE   INTEGRAL 

234.  Integral  the  Limit  of  a  Sum.  An  integral  may  be  regarded 
and  defined  as  the  limit  of  a  sum  of  a  series  of  terms,  and  it  is  in 
this  form  that  integration  is  most  readily  applied  to  practical 
problems. 

235.  Area  of  curve  the  limit  of  a  sum  of  rectangles.  Let  it  be  re- 
quired to  find  the  area  PABQ  included  between  the  given  curve  OS, 
the  axis  of  X,  and  the  ordi- 

nates  AP  and  BQ. 

Let  y  =  x^  be  the  equa- 
tion of  the  given  curve. 

Let  OA  =  a,  and  OB=b. 

Suppose  AB  divided  into 
n  equal  parts  (in  the  figure, 
n  =  6),  and  let  Ax  denote 
one  of  the  equal  parts,  AAi, 

Then  AB  =b  —  a  =  n  Ax. 
At  Ai,  A.^,  •••,   draw  the 
ordinates  AiP^,  A2P2,  •••,  and  complete  the  rectangles  PA^,  P1A2, 
From  the  equation  of  the  curve  y  =  x^, 

PA  =  a^,  P,Ai  =  (a  +  Ax)i,  P.A,  =  (a  +  2 Ax)^,  — ,  Q-B  =  bK 


Area  of  rectangle  PA^  =  PA  x  AAi  =  a^Ax. 


Area  of  rectangle  P1J.2  =  PiA^  X  A^Ao  =  (a  +  Ax)^  Ax. 
Area  of  rectangle  Pg^g  =  Po^o  X  A0A3  =  (a  +  2  Ax)^  Ax. 


Area  of  rectangle  P^B  ■■ 


P,A,xA,B: 
306 


(b  -  Axy  Ax. 


INTEGRATION  AS  A  SUMMATION.  307 

The  sum  of  all  the  n  rectangles  is 

a'  Aaj  +  (a  +  Ax)?  Aa;  -|-  (a  +  2  Ax) i  Ax  H h  (^  -  Ax) J  Ax, 

which  may  be  represented  by  V  x^  Ax, 

where  x^  Ax  represents  each  term  of  the  series,  x  taking  in  succes- 
sion the  values  a,  a  +  Ax,  a  -f  2  Ax,  •••,  h  —  Ax. 

It  is  evident  that  the  area  PABQ  is  the  limit  of  the  sum  of  the 
rectangles,  as  w  increases,  and  Ax  decreases. 

That  is.  Area  PABQ  =  Lim^^^^  ^x^  A.r. 

236.  Definite  Integral.       From  the  preceding  article 

^  x^  Ax  =  a2  Ax  +  (a  +  A.r) '  Ax  +  (a  +  2  Ax) i  Ax  +  •  •  •  +  (6  -  Ax)  i  Ax. 
The  limit  of  this  sum,   as  Ax  approaches  zero,  is   denoted   by 
I    x^  dx.      That  is,  by  definition, 

x-^  dx  =  Lim  ^_^„  V  x^  A.r. 

I    X-  dx  is  called  the  definite  integral,  from  a  to  b,  of  x- f?x. 

It  is  to  be  noticed  that  a  new  definition  is  thus  given  to  the  sym- 
bol I  ,  which    has   been   previously  defined  as  an  an ti -differential. 

The  relation  between  these  two  definitions  will  be  shown  in  the  fol- 
lowing article. 

237.  Evaluation  of  the  Definite  Integral  I    x*  dx.      This  is  effected 

by  finding  a  function  whose  derivative  is  x^. 

,J  =  A^2x_\ 
dx\  3   J 

By  the  definition  of  derivative.  Art.  15, 
dx\S  J  -^  A* 


308  INTEGRAL   CALCULUS 

Hence  3  "^          3  i 

=  x^  +  c, 

A.K 

where  e  is  a  quantity  that  vanishes  with  Ax. 

Hence  -{x  +  Ax}^ ^=x'^  Ax  +  eAx. 

3  o 

Substituting  in  this  equation  successively  for  x, 
a,    a-\-  Ax,    a  +  2  Ax,  •■•,b  —  Ax, 

-  (a  +  Axy  -  ^^  =  a-  Ax  +  cj  Aa;, 
o  o 

O  3  '>  3  1 

-  (a  +  2  x')^  —  -  (a  +  Ao;)^  =  (a  +  A.x')'-^  A.x-  +  e.  Ax, 
o  o 


?  (a  +  3  Aa;)^  -  ?  (a  +  2  Ax)^  =(a  +  2  Ax)^  Ax  +  £3  ^x, 
o  o 


2_^  _  ?  (6  _  A.i;)t  =  (6  -  Ax)  ^ Ax-  +  e„  Aa;. 
o        o 

Adding  and  cancelling  terms  in  the  first  members, 

2&2  _  2_a_2  ^  ^i  ^^  _^  ^^^  _j_  ^^^^^-^_^,  _^  ^^,  _^  2  A;r)2  A.i-  +...+(&_  Aa;)^A.T 
o  o 

+  ci  AaJ  +  to  Ax  +  en,  Ax  +  •  •  •  +  e„  Ax 

=  ;5/'^-^'+2^^'^' ;,  (^) 

Comparing  with  the  figure,  Art.  235,  ^  x-  Ax,  as  we  have  before 

seen,  represents  the  sum  of  the  rj&ctangles,  and   ^  e  Ax  represents 

the  sum  of  the  triangular-shaped  areas  between  these  rectangles  and 
the  curve. 

The  latter  sum  approaches  the  limit  zero,  as  Ax  approaches  zero. 


INTEGRATION   AS   A   SUMMATION  309 

For  if  Ci  IS  the  greatest  of  the  quantities  e„  co,  ...c„,  it  follows  that 
2  £  Ax  <  e^^  A.X', 

that  is,  2^  e  Ax  <  ei,(b  —  a). 

As  e^  vanishes  with  A.r, 


Taking  the  limit  of  (1) 

n 

3 


—  =  Lim^,^o2^_^-'-*Aa!, 


X'  dx  =  tliL  _  ^iL  =  Area  P^iBQ. 
o  3 

Thus  the  value  of  j    .t-^  dx  is  found  from  the  integral 


I  a*-r/.f: 


2£2 

3    ' 


by  substituting  for  x,  b  and  a  in  succession,  thus  giving 
2  6'      2  J 


3  3 

The  process  may  be  expressed 

X''    I 
x'^dx 


2^r     2^_2a^ 
3    U       3  3    ' 

This  is  called  inlof/ratinf/  bettceen  limitfi,  the  initial  value  a  of  the 
variable  being  the  lotcer  limit,  and  the  final  value  b  the  upper  limit. 
In  contradistinction 


is  called  the  indefinite  inteyral  of  a;-(?j;. 


310  INTEGRAL  CALCULUS 

238.  General  Definition  of  Definite  Integral.  In  general  if /(a;)  is  a 
given  function  of  x  which  is  continuous  from  a  to  b,  inclusive,  the 
definite  integral 

r  / (x)dx  =  LimAa-=o  f(a)^x  +f(a  +  Aa;) Aa;  +/(«  +  2  Ax)Ax 

+  •••  +/(&-Aa-)AJ. 

If  j  / (x)  dx  =  F  (a;),     the  indefinite  in tegral, 

£f{x)dx  =  F(b)-F(a) (1) 

This  may  be  illustrated  by  the  area  bounded  by  a  curve  as  in  Art. 
235,  by  supposing  y=f(x)  to  be  the  equation  of  the  curve  OS. 

The  proof  of  Art.  237  may  be  similarly  generalized  by  substitut- 

ing/(x)  for  x'^,  and  F(x)  for  ^^  • 

3  ^h 

Geometrically  the  definite  integral     I   f(x)dx  denotes   the  area 

swept  over  by  the  ordinate  of  a  point  of  the  curve  y=f(x),  as  x 
varies  from  a  to  b. 

It  is  to  be  noticed  that  in  Art.  192,  by  a  somewliat  different  course 
of  reasoning,  we  have  arrived  at  the  same  result, 

Area  PABQ  =  F(b)  -  F(a). 

239.  Constant  of  Integration.  It  is  to  be  noticed  that  the  arbitrary 
constant  C  in  tlie  indefinite  integral  disappears  in  the  definite^ 
integral.  r"   ^ 

Thus,  if  in  evaluating    I    x^  dx,  we  take  for  the  indefinite  integral 

we  find         r..J&=?A'  +  C.    '2<.«.^A     2i=     -.J 

%ya  ,3 


3     '  V  3  ;        3         3 

Or  if  f/  (x)  dx  =  F{x)  +  C, 

jy{x)dx  ^  F(b)  +  C-  [F{a)  +  C]=F{b)-F{d). 


^ 


INTEGRATION   AS   A   SUMMATION 


EXAMPLES 


1.    Compute  V  a;- Aa;  for  different  values  of  Ax. 
When  Ax  =  .2, 

X^^  —2—0—0 

Z^y?  Ax  =  (1-  +  Dr  +  1.4'  + 1^ 

+  178')(.2)  =  2.04. 
When  Ax=.l, 
2^^x-Ax  =  (1^  +  i.r  +  1.2'  +  •  •  • 

+  ]:9')(.1)=2.18. 


When 


Ax  =  .05,   ^x'^x  =  2.2iS. 


dx  =  ~\ 


2.33. 


Curve  OS,  y=or.    0A=1,  GB=2.       0 
Area  PABQ  =  2.33  square  units. 

2.   Compute  V  — ^  for  different  values  of  Ax, 

"^1  X 

When  Ax  =  l, 

When  Ax  =  .5,     V'—  =  1 .503 
•     -^^i  X 

WhenAx=.l,   V'— =  1.42 


Y 

/ 

/ 

/ 

/ 

/ 

J 

B        X 


312  INTEGRAL   CALCULUS 

Lim^^oT'— =  f— =loga-r  =  log4-logl-log4  =  1.386. 

Curve  EQ,  y  =  --     OA^l,  0B  =  4..     Area  PABQ  =  1.386  square 
units. 

3.   Compute  ^  x  Ax,  when  Aa;  =  1 ;  when  Ax  =  .5 ;  when  Ax  =  .2. 

^  Ans.    18;  19;  19.6. 

Find  Lim.j.^oy!  ^^Ax-.  Ans.    20. 

, ;  4.   Compute  V   ^,,     when    Aa:  =  .2;     when   A.r=.l;     Avhen 

^^   Ax=.05.  ol  +  -t-  ^,^g_    .833;  .810;  .798. 

Lim^^T  - — ^-  Ans.   ^=.785. 

■^^0 1  +  X-  4 


'] 


Find 


5.  Computed    logio-^Aa-,    when    A.r  =  l;    when   A.r^.S;     when 
Ax=.3.  '"  Ans.    3.121;  3.150;  3.161. 

Find  Lim^;^  >     logio^jAa;.     J»s.  13  logiol3  — 3  logioe  — 10=3.177. 

K 

6.  Compute  ^  tan  (^  A^,  when  A<^  =5°=^;  when  A(ji  =  ^t:!  when 

^*^^iio'  '  •  ^''■'-  •^^^'  •^^^'  ■^'^^• 

Find  Lim^^r=o  ^  tan  <^  A</).  ^ws.  ■  log^  VS  =  .346. 

i" 

7.     I    (cc^— 4)-a;da;  = 8.     I    —  =2. 

^^«   Va-  —  'f^  Jl  ar^  —  X  +  1      3  V3 

11.     f_i!!*L  =  tan->e^-^.   J    12.     r^-^^  =  2.«^ 
Jo  e-^  +  1  4  Jo    .r  +  4a- 


INTEGRATION   AS  A  SUMMATION'  31;J 

/13.     f  ^-^  A  . 

Ja   V(.i;-2)(3-.x-)  14.     )     ii\n'<f>,l<f>  =  ^. 

15.  j;%i„=.,M.X-l.         jg    r,e,...,<„„  =  r_Lvi. 

12  ^0  4        16 

nr"  COS  26  —  cos  2  rt  ,^      1    ,  TT 
•     I cW  =  l  +-  cos  a. 

'^  ^0      cos  $  —  cos  a  3 

/    19.     f — =  -log(2e). 

20.  I     X-  sin  a;  cZo;  =  tt  —  2. 

21 .  ('"x  log  (.r  +  a)  dx  =  ^\og  (.3  a)  -  ^'. 

22.  (    tan-^-f?.r  =  7r-log4. 
Ju  4 

■   J./   (x-2  +  a^(af'  +  6^')  ~a  +  6V4 


.     _i  rt 

tan  '  - 

b 


By  (5)  and  (6),  Art.  223,  we  find' 

X  sin" xclx=  "  ~      I    s[n"~^xdx, 
n    J» 

n  IT 

I    cos" a; O.J- =  — I    cos"  -a; rtx; 

Jo  n    Jo 

from  which  derive  the  following  results: 


314  INTEGRAL  CALCULUS 

,  24.    If  n  is  even, 


Cos"  X  ax  =  — ^^ ^  -  • 

2.4.6-n       2 


25.   If  w  is  odd, 

r^  •  „   7     r^   «  ^    2. 4. 6. ..(7^-1) 

I    sin"a;f?x=  I    cos"a;aa;  =  —         ^^ ^• 

I/O  »/o  3  •  5  •  7  •  •  •  H 

240.    Sign  of  Definite  Integral.      In  considering  the  definite   inte- 
gral  I   /  (x)  dx,  we  have  supposed  a<b,  and  /  (x)  to  be  positive  be- 

,tween  the  limits  a  and  b.  ^ 

Iif(x)  is  negative  from  x=  a  to  a;  =  6,  A  /(.^')  ^x,  being  the  sum 

of    a   series    of    negative    terras,    is    negative,    and    consequently 

Xf  (x)  dx  is  negative. 

If  /(x)  changes  sign  between  x  =  a  and  x=b,   I  f(x)  dx  is   the 
algebraic  sum  of  a  positive  and  a  negative  quantity. 


For  exam 


r 


cos  X  dx—l  =  area  A  OB. 


cos  xdx  =  —  2  —  area  BCD. 


INTEGRATION   AS   A   SUMMATION  315 

Sn 

I      COS  X  dx=  —1  =  1  —  2. 

r  "  COS  X  dc  =  0  =  1  —  2  +  1. 
The  change  of  sign  resulting  from  a  <  b  is  considered  in  Art.  243. 

241.  Infinite  Limits.  In  the  deiinition  of  a  definite  integral  the 
limits  have  been  assumed  to  be  finite.  When  one  of  the  limits  is 
infinite,  the  expression  may  be  thus  defined  : 

J    f{x)  dx  =  Lim^^,    i  f(x)  dx. 
For  example,  consider  Ex.  12,  p.  312, 

Jo    ar  +  4a^  Jo  or +  4  a-  \  JaJ 

Referring  to  Art.  126,  we  find  the  geometrical  interpretation  of 
this  result. 

The  area  included  between  the  curve,  the  axes  of  X  and  Y,  and  a 
variable  ordinate,  approaches  the  limit  27rcr,  as  the  distance  of  the 
ordinate  is  indefinitely  increased. 

-^,  we  find 

1       X 

—  =  Linifc^^  log  6  =  Qo  . 

1       X 

—  has  no 
1      ^ 
meaning. 

242.  Infinite  Values  of /(.r).     In  the  definition  of  J   /(x)cte, /(a-) 

is  assumed  to  be  a  continuous  function  from  x  =  a  to  x  =  b.  If 
f(x)  is  continuous  for  all  values  from  a  to  6  except  x=  a,  where  it 
is  infinite,  the  definite  integral  may  be  defined  thus : 


Ja+h 


316  INTEGRAL   CALCULUS 

If/(/;)  =00,  /(x-)  being  continuous  for  other  values  of  x, 


For  example,  con 

sider  Ex.  9,  p.  312, 

r      dy       . 

Here        ^ 

CO ,  when  y  —  a. 

Va^-/ 

Hence 

"      "^y       -Lim 

2 

"      dy           X  •         /'  • 

-^ =  Lim„_n   sin~ 

a 

-i) 

l^a^-f 

TT          TT 

~2~6 

TT 

~3' 

Another  example 

is  Ex 

s: 

13,  p.  313, 
dx 

^/{x-2){3-x) 

Here  — 

=  00,  when  x  =  2, 

and  also  when 

x  =  3. 

-^(x-2)(3-x) 

„  C^  dx  -r.  r-"  dx 

Hence      I    — ^^^=^^=z=  =  Lim^^o  I       —  

J2  V(.x-2)(3-ic)  •^-+*  V(a;-2)(3-a;) 

=  Lim^o  sin-'  (2  x  -  5)  T  '=  Lim^^o  [sin"'  (1-2  h)  -  sin"'  {-1  +  2  Ji)'] 


sin  '1  —  sin  '(—  1): 


If  f(x)  is  infinite  for  some  value  c  between  a  and  b,  and  is  con- 
tinuous for  other  values,  the  definite  integral  should  be  separated 
into  two. 

rf{x)dx=  rf{x)dx+  C''f(x)dx.     See  Art.  243. 
These  new  definite  integrals  may  be  treated  as  already  explained. 


INTEGRATION   AS   A   Sl'MMATlON  317 

EXAMPLES 

/     '   Jo    (l+a-)-"~4'  ■   Jo    (.r  +  a=')(a;-  +  6-)~2o6(a  +  6)* 

^'     r^4^  =  log(2  +  V3).  4.     T— ^    =llog2. 

^1    y'.^i'  _  1  c/i    a;  (1  +  a;  )      J 

243.  Change  of  Limits.  The  sign  of  a  definite  integral  is  changed 
by  the  transposition  of  the  limits, 

f"f{.v)  dx  =-  C  '/(x)  dx. 

\Jlj  « 'a 

This  is  evident  from  (1),  Art.  238,  and  also  from  the  definition. 
For  if  X  varies  from  h  to  «,  the  sign  of  A.i-  is  opposite  to  that  wiiere  x 
varies  froni  a  to  6.  Hence  the  signs  of  all  the  terms  of  ^^J'(-^")  ^^ 
will  be  changed,  if  the  limits  a  and  h  are  transposed. 

Hence  f  /  (a-)  dx  =  -  Cf  (x)  dx. 

A  definite  integral  may  be  separated  into  two  or  more  definite 
integrals  by  the  relation, 

J /(x)  dx  =  jT /(.^O  dx  +  J* /(.'•)  dx. 
This  follows  directly  from  the  definition. 

244.  Change  of  Limits  for  a  Change  of  Variable.  When  a  new 
variable  is  nsed  in  obtaining  the  indefinite  integral,  we  may  avoid 
returning  to  the  original  variable,  by  changing  the  limits  to  corre- 

^  spond  with  the  new  variable. 
For  example,  to  evaluate 

dx  


-'o    1  + 


Vx 


Vx  =  z. 


318  INTEGRAL  CALCULUS 

Then  we  have  ^^  _^2zdz^ 

1+^X        1  +  2! 

Now  when   a;  =  4,  2;  =  2  ;    and  when    a;  =  0,    2  =  0. 
Hence         C-^=  f  2_1^  =  2  [._  log(l +.)]f 
=  4-21og3. 

EXAMPLES 

1.  Cx-J^^+2dx  =  ^.  Let    x  +  2  =  z\ 

2.  C\x-2yxdx=   .^"^^^^    .  Let    x-2  =  z. 
./2                             7r  +  3w  +  2 

3      f  ^'^^^    ,^^g(.3  +  ^5).  Let    x'  +  l=z^ 


^-   J^  V2  ax  —  X-  dx  ■■ 


Let    x—  a  —  a  sin ^. 


^-   J,  V(a;-a)(&-a;)d'c  =  J  (p-af.  Let    a;  =  a  cos- <^  +  &  sin^  </,, 


7.    jrV-^^)^c?a;==^^'. 


Let    a:  =  a  sin^  9. 


^~  V64      48/ 


INTEGRATION    AS   A   SUMiMATION  819 

245.    Definite  Integral  as  a  Sum.     In  the  application  of  integration 
it  is  often  convenient,  in  forming  tlie  definite  integral  from  the  data 

of  the  problem,  to  regard  i   f{x)  dx  as  the  sum  of  an  infinite  number 

of   infinitely  small  terms,  f{x)dx  being  called  an   element  of  the 
required  definite  integral. 
From  this  point  of  view, 

Cf(x)  dx  =f(a)  dx  +  f(a  +  dx)  dx+f(a  +  2  dx)  dx  +   -  +f{b)  dx. 

This  may  be  regarded  as  an   abbreviation  of  the  definition  of  a 
definite  integral  given  in  Art.  238. 


CHAPTER   XXVIII 

APPLICATION   OF   INTEGRATION   TO    PLANE   CURVES. 
APPLICATION    TO    CERTAIN   VOLUMES 

246.  Areas  of  Curves.  Rectangular  Coordinates.  We  have  already 
used  this  problem  as  an  illustration  of  a  definite  integral.  We  will 
now  consider  it  more  generally,  and  derive  the  formula  for  the  area 
in  rectansfular  coordinates. 


247.  To  find  the  area  between  a  given  curve,  the  axis  of  X,  and  two 
given  ordinates  AP  and  BQ ;  that  is,  to  find  the  area  generated  by  the 
ordinate  moving  from  AP  to 
BQ. 

Let  OA  =  a,  OB  =  b. 

Let  X  and  y  be  the  coordi- 
nates of  any  point  P.,  of  the 
curve ;  then 


X  +  Ax,   y  +  Ay, 


will  be  the 


L'dinates  of  P.,, 


The   area  of   the  rectanj 
P,A,A,  is 

P.,Ao  X  A.A^  =  y  A.T.* 


Ai     Ao    A,    A4 


The  sum  of  all  the  rectangles  PAA^,  P^A^A^,  P-iAui^,  •••,  maybe 
represented  by  2a^'/^^'- 

The  required  area  PQBA  is  the  limit  of  the  sum  of  the  rectangles, 
as  Ax  is  indefinitely  diminished.     That  is 


A 


=£ydx, 


liy  Art,  245,  one  readily  sees  that  this  rectangle  is  an  element  of  area. 
320 


APPLICATION   OF   INTKGllATIOX   TO    PLANE  CLRVKS     8:il 


the  lower  limit  a  =  OA,  being-  the  initial  value  of  x,  and  the  upper 
limit  b=  OB,  the  final  value  of  x. 

Similarly  the  area  between  the  curve,  the  axis  of    Y,  and  two 
given  abscissas,  GP  and  IIQ,  is 


A 


J 'ft 
xdy, 


the  limits  of   integration  being  the  initial   and   final  values  of  y, 
cj  =  OG,   and  h  =  OU. 

EXAMPLES 

1.    Find  the  area  between  the  parabola  y-  =  Aax  and  the  axis  of 
X,  from  the  origin  to  the  ordinate  at  the  point  (li,  k). 


Here  A=jif(lx=r2ahhh 


_4aM  * 

4 

ch 

^           0 

o 

Since 

Jc--- 

=  \ah,  A-  = 

--2  a 

'l'\ 

which 

gives 

A^-h2 ci^h^  =  =  hk  =  -  OMPN. 
o  3  3 


2.   Find  the  area  of  the  ellipse 


a-     0- 


Area  BOA 

I    yclx  =  -  \     Va-  —  .r  fix 

hVx    /—, r,  ,  a-  ■    _-iX~Y 

=  -    -  Va-  —  X-  +  —  sm  >  - 
a|_2  2  ajo 


_  irah 
~~    4 

*  In  finding  areas,  after  \\\e  element  of  area  and  the  limits  of  inteyration  are 
chosen,  the  problem  becomes  purely  mechanical. 


322  INTEGRAL   CALCULUS 

The  entire  area  =  irah. 

Or  we  may  integrate  by  letting  x  =  a  sin  <^. 


Then    rVa'-iB2da;  =  a2  rcos-</>f/(^=-  fVl +cos2d,)dcf,=  ^*. 
»^  Jo  2  Jo  ^■'    ^        A 


a     4         4 


3.    Find  the  area  included  between  the  parabola  j;-  =  4  caj,  and  the 

witch  y  =  _,A^^ .  ^„,;  (2\  _  i] a--. 

X-  +  4:  a-  \^  sj 

Having  found  the  point  of  intersection  P,  (2  a,  a),  we  proceed  as 
follows : 


0        2a       M  X 

Aiea,*AOF=  AOMP-  OMP* 

—  I      ~^ : — :;  ~^   I      "  =  ■"■«" • 

Jo     x'  +  4a-     Jo      4  a  3 

Area  between  two  curves  =  f  2  tt  —  -\a^. 

4.    Find  the  area  of  the  parabola  ^ 

(y-5)2  =  8(2-ar), 
on  the  right  of  the  axis  of  F.  Ana.    10%. 

*  Length  of  element  of  area  is  the  y  of  the  witch  minus  the  y  of  the  parabola. 


I 


APPLICATION    OF   INTEGRATION    TO    PLANE   CLUVES      323 

5.  Show  that  the  area  of  a  sector  of  the  equilateral  hy[>erbola 
01?  —  ■jf  =.  or,  included  between  the  axis  of  X  and  a  diameter  through 

the  point  {x,  y)  of  the  curve,  is    ^log'       ■' » 

6.  Find  the  entire  area  within  the  curve  (Art.  133)  (-)+(-)   =  !• 

Ans.    '-  -n-ab. 
4 

7.  Find    the    entire    area    within    the    hypocycloid    (Art.    132) 

rf.?,  -)-  y*=  a^-     Let  X  =  a  sin^  (f>.  Ans.  ^-^  . 

o 

8.  Find  the  entire  area  between  the  cissoid  (Art.  12/5)  ?/-  = 

2  a  —  X 

and  the  line  x  =  2  a,' its  asymptote.  Ans.   3  -n-d-. 


9.    Find  the  area  of  one  loop  of  the  curve  (Art.  134)  a*y-  =  arx*  —  a^. 


,         (it  ,   V3\o2 

10.  Find    the    area    of    the    evolute    of   the    ellipse    (Art.   167) 

11,  What  is  the  ratio  between  a  and  h,  when  the  areas  of  the 
ellipse  and  its  evolute  are  equal  ? 

.       a      V5  +  V2      „.. 
Ans.  -  = ' =  Z.ll. 

^  V3 


12.    Find  the  area  included  between  the  parabolas 

?/-  =  ax     and     .i*^  =  by.  Ayis.  — - . 


^■:) 


324  INTE(;ilAL   CALCULUS 

13.  Find  the  area  included  between  the  parabola 

y"'  =  'lx    and  the  circle    y'^  =  ^x  —  o?.  An. 

14.  Find  the  area  included  between  the  parabola 

2/^  =  4  ax,   and  its  evolute  (Art,  167)  27  ay"-  =  4  (x  —  2  a)^. 

.       352  V2  2 

Ans. — a  . 

15 

Parametric  Equations.  Instead  of  a  single  equation  between  x  and  y 
for  the  equation  of  a  curve,  the  relation  between  x  and  y  may  be  ex- 
pressed by  means  of  a  third  variable.     Thus  the  equations 

X  =  a  sin  ^,       y  =  a  cos  <^, (1) 

represent  a  circle;  for  if  wo  eliminate  <^  from  (1)  Ave  have 
a;-  4-  ?/-  =  a-  (sin-  ^  +  cos'-  </>;  =  o". 

Equations  (1)  are  called  the  imrametric  equations  of  the  ciMe,  and 
the  third  variable  <^  is  called  the  parameter. 

The  formula     A=  i  y dx  is  applied  to  (1)  by  substituting 
y  =  a  cos  cf),    dx  =  a  cos  cf)  d<j>. 

For  a  quadrant  of  the  circle 

A  =  \    y  dx  =3  I    cr  cos-  ^  «^  =  -— . 
Ja  Ji)  4 

15.  Find  the  area  of  one  arch  of  the  cycloid 

X  =  a{d  —  sin  6),     y  =  a  (1  —  cos  6).  Ans.  3  -n-a-. 

16.  The  parametric  equations  of  the  trochoid,  described  by  a  point 
at  distance  b  from  the  centre  of  a  circle,  radius  a,  which  rolls  upon  a 
straight  line,  are 

X  =  ad  -—  h  sin  6,     y  =  a  —  h  cos  6. 

Find  the  ai-ea  of  one  jircli  of  the  trochoid  above  the  tangent  at  the 
lowest  points  oT  tlic;  curve. 

Ans.  7J-  (2  a  +  h)  h,  when  h  <a  ox  l>>  a. 


APPLICATION   OF    INTE(; RATION   TO   PLANE   CURVES      325 

248.    Areas  of   Curves.     Polar  Coordinates.     To  find  the  Area  POQ 
included  between  a  Given  Curve  PQ  and  Two  Given  Radii  Vectores ;  that 
is,  to  find  the  area  generated  by  the 
radius  vector  turning  from  OP  to 
OQ. 

Let     POX=a,    QOX  =  ft. 

Let  r  and  0  be  the  coordinates  of 
any  point  P«  of  the  curve,  then 

r  +  Ar,    e  +  AO, 

will  be  the  coordinates  of  P^. 

The  area  of   the  circular  sector 
P2OR,  is 

i  OP.  xP,Ro  =  \r-rAe  =  \  r  AO. 


The  sum  of  the  sectors  POP,  P^OR^,  P^OR^, 
resented  by  s  1 


may  be  rep- 


A^. 


The  required  area  POQ  is  the  limit  of  the  sum  of  the  sectors, 
as  A^  approaches  zero.     That  is, 

•/3 


in 

2Ja 


7-'  (W, 


the   initial   value   of  9,   a=  POX,  being  the  lower  limit,  and  the 
final  value  oi  0,   jS=  QOX,  the  upper  limit. 


EXAMPLES 
1.   Find  the  area  of  one  loop  of  the  curve  (Art.  144)    r  =  a  sin  2  6. 

A  =  lf  "r  dO  =  I  fa-  sin^  2  9(10  =  '-^'  f\l  -  cos  4  0)  cW 

2»/(j  2«/u  4  *yo 

aY„     sin4«Y     ^a' 


326  INTEGRAL   CALCULUS     • 

The  entire  area  of  the  four  loops  =  ^', 
which  is  half  the  area  of  the  circumscribed  circle. 

2.  Find  the  entire  area  of  the  circle  (Art.  135)    r  =  a  sin  6.  „ 

Ans.  — —. 
4 

In  the  two  following  curves  find  the  area  described  by  the  radius 
vector  in  moving  from  ^  =  0  to  ^  =  t- 

3.  r  =  sec^  +  tan^.  ^1"^^-    V'--^- 

4.  r  =  a(l-tan2^).  j_ns.  [^^-'\i\ 

5.  Find  the  entire  area  of  tlu^  cardioid  (.\rt.  141)  r  =  a(l  —  cos  0). 


Also  find  the  area  from  ^  =  ^  to  ^  =  •-^.  Ans.  (3  tt  -  2)  - ^ 

4  4  b. 

6.  Fuul  tlie  ariMX  (lescrilxMl  liv  f  he  i-ndhis  voctor  in  the  ])aralHil;i 
(Art.  139)  r  =  <i  M-(-_^.  In.ni  6  =  ^)  to  0  =  ^-  Ai,x.    '.;'"• 

Also  find  the  area  from  6  =  ^  to  6  =  —^.  Ans.  

3  3  9  Vo' 

7.  Find  the  entire  area  of  the  lemniscate  (Art.  143)  ?-^  =  a^  cos  2  6. 

Ans.  a-. 

8.  Show  that  the  area  bounded  by  any  two  radii  vectores  of  th(> 
reciprocal  spiral  (Art.  137)  r6  =  a  is  proportional  to  the  difference 
between  the  lengths  of  these  radii. 


9.    In  the  spiral  of  Archimedes  (Art.  136),  r  =  ct9,  find  the  are 
described  by  the  radius  vector  in  one  entire  revolution  from  ^  =  0. 

Ans.  *''"•- 


3 

Also  find  the  area  of  the  strip  added  by  the  wth  revolution. 

Ans.  8(«-l)7rV. 


APPLICATION  OF   INTEGRATION   TO   PLANK   CURVES      327 


10.  Vhul  tlie  area  of  the  part  of  tlie  circle  (Art.  l.'>5) 

)•  =  a  sill  ^  +  ^  cos  0,     from  ^  =  0  to  ^  =  -. 

2 

Ans.   !l(^+J^  +  ^^ 
8  2 

11.  Find  the  area  connnon  to  the  two  circles  (Art.  13;")\ 

)•  =  a  sin  6  +  b  cos  6,     r  =  a  cos  6  +  h  sin  6. 

Ans.   f-  +  2tan-^^V^+^'-^-^-'^   wherea>?,. 
\2  ay      4  4 

12.  Find  the  area  of  the  loop  of  the  Folium  of  Descartes  (Art.  127) 


13.    Show  that  the   lii 


g  tan  0  sec 
1  +  tan'^  d 

2  a  sec  0 


Ahs. 


So'' 


(x  +  y  =2a),  divides  the 
1  +  tan  6' 

area  of  the  loop  of  the  preceding  example  in  the  ratio  2:1. 

a 
14.    Find  the  entire  area  within  the  curve  (Art.  145)  r  =  a  sin^  - ,  no 


part  being  counted  twice 


A>,s.   (IOtt  +  ^^Vs)  ^. 


249.   Lengths    of    Curves.     Rectangular    Coordinates.     To   find   the 
Length  of  the  Arc  FQ  between  Two  Given  Points  P  and  Q. 

Let    OA^a,    OB=h. 

Denoting  the  required  length   of 
arc  by  s,  Ave  have  from  (1),  Art.  l.jo, 


ds 


yjl  +  f'^)\lx. 


dx 


Hence 
s 


and  between  the  given  limits 


X 


\^+(IT-' 


(1)  0 

the  limits  being  the  initial  and  final  values  of  x. 


B     X 


328  INTEGRAL   CALCULUS 

We  may  also  use  the  formula 


the  limits  being  the  initial  and  final  values  of  y, 
g  =  OG,  and  h  =  OH. 

EXAMPLES 

1.    Find  the  length  of  the  arc  of  the  parabola  ?/^  =  4aa:,  from  the 
vertex  to  the  extremity  of  the  latus  rectum, 

XT  f^?/      «^, 

Here  -7"  =  — ;' 

ax     ^i 

therefore  s  =fj(^  +  ff  dx  =  jTY^^'V"  dx. 
This  may  be  integrated  by  Ex.  13,  p.  305,  making  6  =  0. 
rfn_+x\h  ^^  ^  V  ax  +  X-  +  a  log  ( V  a  +  a;  +  V^) 
rra±x\h  ^^^  ^  ^  j-^2  +  log  (1  +  V2)]  =2.29558 a. 
Or  vs^e  may  use  the  formula  (2), 


=rvi+(^)#. 


4  a'     dy      2  a 


=  ^\l Vf  +  U,:'  +  ^' log  (y  +  v7T4^)]^" 
=  a[V2  +  log(l+ V2)] 


APPLICATION   OF   IN  rEGIlA  TlOX    TO   PLANE   CURVES      329 

/  2.    Find    the    length  of    the   arc  of   the    semicubical    parabola 

(Art.  130)  af-  =  x\  from  x  =  -  to  x  =  ba.  Ans    ^ 

^  8  ■ 

3.    Find  the  entire  length  of  the  arc  of  the  hypocycloid  (Art.  132) 
x^  +  y^  =  a^-  Ans.  6  a. 

/   4.    Find  the  length  of  the  arc  of  the  catenai-y  (Art.  128) 

a    ? 
2/  =  -(e«  +  e  "), 

from  .^•  =  0  to  the  point  {x,  y).  Ans.   ^  (e"  —  e  "). 


5.   Find  the  length  of  the  arc  of  the  curve 

y  =  log  sec  X,   from  a;  =  0  to  .x  =  -. 

Ans.  log  (2  +  VS). 
y-  6.    Find  the  length  of  the  curve 

6  xy  =  x'^  +  3,  from  a-  =  1  to  x*  =  2.  Ans.   -y. 

7.   Find  the  perimeter  of  the  loop  of  the  curve 

9  ay^  =  (x  —  2  a)  {x  —  o  a)-.  Ans.  4  V3  a. 

^  8.    Find  the  length  of  that  part  oi  the  evolute  of  the  parabola 
(Art.  167)  27  ay-  =  4  (a*  —  2  of  included  within  the  parabola  ?/^  =  4  ax. 

Ans.  4  (3  V3  -  1)  a. 
9.   Find  the  length  of  the  curve 

y  =  log      ~    ,   from  ar  =  1  to  ;r  =  2. 

"^  <-''  +  1  Ans.  log  (e  +  e"^). 


^  10.   Find  the  length  of  one  quadrant  of  the  curve  /  -  |   +  [  -  )   =  1- 

Ans.   -^^-i= ■ — . 

(I  +  b 

11.    The  parametric  equations  of  a  curve  are  x  =  e^  s\u'$,  y  =  e"  cos  6. 
Find  the  length  of  arc  from  e  =  Otod  =  l.  A,is.  V2  (e?  - 1). 


330 


INTECaiAL   CALCULUS 


12.    The  parametric  equations  of  the  epicydoid,  the  radius  of  the 
fixed  circle  beiuy  a,  and  that  of  tlie  rolling  circle  -,  are  (Art.  131) 

x  =  -(3  cos  (f>  —  cos  3  (fi), 

7/  =  ^(3sin<^-sin3<^), 

<^  being  the  angle  of  the  fixed  circle,  over  which  the  small  circle  has 
rolled. 

Find  the  entire  length  of  the  curve.  A71S.  12  a. 

250.    Lengths  of  Curves.      Polar  Coordinates.      To  find  the  Length 
of  the  Arc  PQ  betvi^een  Two  Given  Points  P  and  Q. 
Let      POX^a,  Q0X  =  /3. 
We  have  from  (3),  Art.  156, 


ds 

therefore 


=[-(i)7 


fW: 


the  limits  being  the  limiting  values 

of  e. 

Or  we  have     rls  =  \  1  +  r-( — j    r  dr ; 
therefore  s  =  fTl  +  ''•'(-)']'  <^' 


(2),  Art.  156, 


(2) 


the  limits  being  the  limiting  values  of  r.      That  is,  OP  — a,  OQ  =  b. 


EXAMPLES 


1.   Find    the    length    of    the    arc   of   the    spiral   of   Archimedes 
(Art.  136),  r  =«0,  from  the  origin  to  the  end  of  the  first  revolution. 


Here 


—  =  a,  and  we  have  by  (1), 
da 


APPLICATION    OF    INTKiniATlON    TO    PLAXK    CLKVKS      331 

s  =  f  {a'^  +  a-) '  dO  =  a  f  (1  +  $') '  dd 

rO^/YTJ^  ,  1 


L+^'  +  |iog(^  +  Vi+^r)T 


=  a    ttVI  +  47r-  +;^log  (2  TT  +  Vl  +  47r-)~|. 
Or  we  ina}'  use  the  formula  (2) 


j: 

-a       1 

■\1  + 

tfj 

Ir. 

a 

_1 
tt 

V  a-  a  Jo 

=  ?^  P'  Vr-+  cr  +  ^'log  (r  +  V/"  +  a-jT" 

=  a  L  V4  7r-  +  1  +  p^  log(2  TT  +  V4  7r-  +  1)  . 

2.    Find  the  entire  length  of  the  circle  (Art.  135)  r  =  2  a  sin  6. 

Ans.  2  tto. 

'  3.    Find  the  length  of  the  arc  of  the  circle  (Art.  135) 

r  =  a  sin  9  +  b  cos  6,   from  ^  =  0  to  (r,  6).  Ans.  ^^a-  +  b'  0. 

v4.   Find  the  length  of  the  logarithinic  s]»irrd  {.\rt.  1.").^  r  =  (■■'",  from 
the  point  {)\,  6^  to  {u,  dS),  using  the  formula  (2),  and  the  equation 


332  INTEGRAL   CALCULUS 

^  5.   Find  the  entire  length  of  the  cardioid  (Art.  141) 

r=a(l  — cos^).  Ans.%a. 

Also  show  that  the  arc  of  the  upper  half  of  the  curve  is  bisected  by 

6.  Solve  Ex.  5  by  using  formula  (2)  and  the  equation  Q  =  vers"'-. 

7.  Find  the  arc  of  the  reciprocal  spiral  (Art.  137)  rQ  =  a,  from 

6 
'  8.   Find  the  arc  of  the  parabola  (Art.  139)  r  =  a  sec-  -  from 

^  =  0  to  ^  =  |.  Ans.  fsec  -  +  log  tan  ^V- 

9.    Find  the  entire  length  of  the  arc  of  the  curve  (Art.  145) 

.   .,0  ^       Sira 

o  ^ 

Also  show  that  the  arc  AB  is  one  third  of  OABC. 
Hence  OA,  AB,  BC,  are  in  arithmetical  progression. 

6 
10.   Find  the  entire  length  of  the  curve  r  =  a  sin"  -,  n  being  a  posi- 
tive integer. 

See  for  integration  Exs.  24,  25,  p.  314. 

2  a,  when  n  as  even. 


1.3-5 --(u-l) 

l.S.5-"n 
2.4.6.- (n-1) 


■n-a,  when  w  is  odd. 


APPLICATION    OF    INTEfJRATlOX    TO    I'l.AXK    ClTlVIvS      33S 

251.   Volumes  of  Surfaces  of  Revolution.     To  find  the  Volume  gener- 
ated by  revolving  about  OX  the  Plane  Area  APi^B. 

Let  OA  =  a,   OB  =  h. 

Let  X  and  y  be  the  coordinates 
of  any  point  P,  of  the  given 
curve. 

It  is  evident  that  the  rectan- 
gle P.,AoA^  will  generate  a  right 
cylinder,  whose  volume  is 

Try-  Ax. 

The  sum  of  all  these  cylinders 
may  be  represented  by 

7r2^  //-  A.r. 

The  required  volume  is  the  limit  of  the  sum  of  the  cylinders,  as 
A.X'  approaches  zero.     That  is, 

F^  =  TT  I    //-  dx. 

Similarly  the  volume  generated  by  revolving  PGIIQ  about  OY  is 

Vy  =  •"■  I   '^''  dy, 
where  0G  =  g,  and  OH  =  h. 


Y 

6                .    q 

) 

F 
R/ 

F 

^ 

/ 

h 

/ 

i     ^ 

/ 

'•> 

Arc 

c 

)               1 

\       A, 

^, 

V. 

h,  f 

[       X 

EXAMPLES 
1.   Find  the  volume  generated  by  revolving  the  ellipse 

about  its  major  axis,  OX.     This  is  called  the  2^rolcUe  spheroid. 

—  =7rl    y2c?.«  =  :rl     —  ( tt- —  j;-) da;  =  — ;p (  a-x- —  —     = — - — 
2        «/o  '  "^0    cr  a-  \  ojo  o 

V=lnab\ 
o 


384 


INTEGRAL   CALCULUS 


2.   Find  the  volume  generated  by  revolving  the  ellipse  about  its 
minor  axis,  OY.     This  is  called  the  oblate  spheroid. 


y=,j\uiy  =  i§j\v-,f)a,j 


2  7ra-b 
3 


V=-7ra'b. 
3 


3.  If  the  parabola  y-  =  4  aa;  is  revolved  about  OX,  show  that  the 
volume  from  a;  =  0  to  a;  =  2  a  is  one  third  the  volume  from  a;  =f  2  a 
to  a;  =  4  a. 

4.  Find  the  volume  generated  by  revolving  the  segment  LOL'  of 
the  parabola  about  the  latus  rectum  LL'. 


Here     l^=  tt  f'^PiV)-  dy  =  tt  C'\a  -  a;)^ 
Jo    V        4  ay     -^      15 


32 

Ans.    —  ira^. 
15 


5.    Find  the  volume  generated  by  revolving 
about  OX  one  loop  of  the  curve  (Art.  134) 


3o 


6.    Find  the  entire  volume  generated  by  revolving  about  OX  the 
hypocycloid  (Art.  132)  x^  +  t/'"  =  aK 


32 
Ans.    — —  ttoP. 
105 


7.    Find  the  volumes   generated  by   revolving   about    OX,  and 
ibout  OY,  the  curve  (Art.  133)  f'^)'  +  (-)  =1- 


39  4      o 

Ans.  Vx  =  ^  7ra?r'.      K,  =  -  7ra-&. 
35  5 


APPLICATION   OF   INTEGRATION    TO    PLANE   CURVES      335 
> 

8.  The  part  of  the  line  ''^'  +  ^=1,  intercepted  between  the  coor- 
dinate axes,  is  revolved  about  the  line  x  =  2  a.     Find  the  included 

volume.  Ans.   -ncrh. 

o 

9.  The  segment  of  the  parabola,  ar'  — 3a;  +  2  ?/  =  0,  above  OX,  is 

revolved  about  OX.     Find  the  volume  generated.  Ans    ^^. 

40 

"^10.  A  segment  of  a  circle  is  revolved  about  a  diameter  parallel  to 
its  chord.  Show  that  the  volume  generated  is  equal  to  that  of  a 
sphere  whose  diameter  is  equal  to  the  chord. 

■<^  11.    Find  the  volume  generated  by  revolving  about  OF  the  witch 
(Art.  126),  y  =  J"^^    ,,  from  (0,  2  a)  toy  =  a.     Ans.  4  (log  4-1)  ttci^ 

12.  Find  the  volume  generated  by  revolving  the  upper  half, 
ABA'OA,  of  the  curve  (Art.  133)  /^^Yi-f^Y  =  1,  about  the  tangent 

g  about  OA'  the  area 

included  between  the  ellipse  '—-\--~  =  1,  and  the  parabola  2  ai/-  =  3  b-.c. 
a-     b-  ^ ,, 

Ans.    -^irab'-. 
48 

X  X 

14.  A  segment  of  the  catenary  (Art.  128),   y  —  j-(e''  +  e  "),  by  a 

chord  through  the  points  .t  =  ±  a  log  2,  is  revolved  about  the  tangent 
at  the  vertex.     Find  the  volume  generated.  .. 

Ans.    3flog2-iij7ra". 

15.  Find  the  volume  generated  by  revolving  about  the  latus  rec- 
tum of  the  ellipse  ■^, -!-•['.,  =  I,  the  segment  cut  off  by  Ihc  latns 
rectum. 


(. 


^     13.    Find  the  volume  generated  by  revolvin 


Ans.    2  Trf  (lb-  —  ab\a-  —  ir  sm  '  -   . 

V  3a  a 


336 


INTEGRAL   CALCULUS 


252.   Derivative  of  Area  of  Surface   of   Revolution.      In   order  to 

obtain  the  formula  for  the  surface  generated  by  the  revolution  of  a 
given  arc,  it  is  necessary  to  find  the  derivative  of  this  surface  with 
respect  to  the  arc. 

Let  S  denote  the  surface  gen- 
erated by  revolving  about  OX 
the  arc  s,  AP. 

Using  for  abbreviation  the 
expression  "Surf  (  )"  to  denote 
"the  surface  generated  by  re- 
volving (  )  about  OX,"  we  have 

S  =  Surf  (s),    A^  =  Surf  (As). 

This  may  be  Avritten 

Surf  (A.s) 


A-S 


Surf  (Chord  FQ) 


Surf  (Chord  PQ) (1) 


Now  the  surface  generated  by  the  chord  PQ  is  the  convex  surface 
of  the  frustum  of  a  right  cone,  which  is  the  product  of  the  slant 
height  by  the  circumference  of  a  section  midway  between  the  bases. 

Hence  Surf  (Chord  PQ)  =  2  tt  f^K±Q±[\  chord  PQ 


2^?l±2l±^ChovdPQ 


=  7r(2  2/  +  A?/)Chord  PQ. 


Substituting  this  for-the  last  factor  in  (1),  and  dividing  both  sides 
by  A.s,  we  have 


A.s  ' 


Surf  (A.s) 


Surf  (Chord  PQ) 


r(2  2/  +  Ay) 


Chord  PQ 

As 


Taking  the  limit  of  each  member,  as  As  approaches  zero,  noticing 
that 


LiniAs= 


Surf  (A.s) 


Surf  (Chord  PQ) 


r 


APPLICATION   OF   INTEGRATION   TO   PLANK   CURVES     337 
and 


,  •           Chord  PQ      .  , 

Lim^.^o =  Ij    we  have 


As 

dS      y.  AS     o 

as  As 

Similarly  if  OY  is  the  axis  of  revolution, 

—  =  2,rx-. 
ds 

253.    Areas  of  Surfaces  of  Revolution.     To  find  the  Area  of  the  Sur- 
face generated  by  revolving  about  OX  the  Arc  FQ. 

By  the  preceding  article  we  have 
dS 


hence 


ds 


■^^y\ 


f'"^ 


ds. 


To  express  this  in  terms  of  x  and 
y,  we  have  from  (1),  Art.  155, 

1^ 


H'^(:m'- 


which  gives 


le  axis  of  rt 


dx.    .    (1) 
If  OF  is  the  axis  of  revolution, 


Or  we  may  use 


of  (2)  i;  =  2,j;.[, +(!)=]'„,. 


and  instead  of  (1)  we  have 


and  instead 


B    X 


(2) 

(1') 
(20 


338  INTEGRAL   CALCULUS 

EXAMPLES 
1.    Fiud  the  area  of  the  surface  generated  by  revolving  about  OX 
tlie  hypocycloid  (Art.  132)     x^  +  ?/^  =  ai 

Here  y  =  (J-  x'^)i,        ^  =.  -  (a?  -x^)^x-\ 

Using  (1)     |^,  =  2^"(a?-^i)t[l+^^J^^ 

Jl^"        2.  Z    ^  ft~3  1    /*"         2  2     3      _  1 

I    (as  —  a;3)2  —  clx  —  2  -n-a^  I    (a^  —  x^)^x  ^dx 

Or  we  may  use  (1')        -^  =  —  (a^  —  ?/3) -^/"s. 

'J  2.  Show  tliat  the  area  of  the  surface  generated  by  revolving  the 
parabola  y"^  =  4  dx,  about  OX,  from  a;  =  0  to  a;  =  3  a,  is  one  eighth  of 
that  from  a;  =  3  a  to  ;i'  =  15  a. 

3.  Find  the  area  of  the  surface  generated  by  revolving  about  OX 
the  loop  of  the  curve   9  caf  —  x(3a—  xf.  Ans.   3  Trcr'. 

4.  Find  the  surface  generated  by  revolving  about  OX,  the  arc 
of  the  curve  6  a^  xy  =  a;*  +  3  a*,  from  a;  =  a  to  a;  =  2  a.  .„ 

Ans.    -— Tral 
16 

5.  The  arc  of  the  preceding  curve  from  a;  =  a  to  a;  =  3  a,  revolves 
about  OF.     What  is  the  surface  generated  ?       Ans.  (20 +  log3)7ra-. 

V     6.    Find  the  surface  generated  by  revolving  about  OF  the  curve 
4  2/  =  a^  —  2  log  a^j  from  a;  =  1  to  a;  =  4.  Ans.  24  tt.     ,, 

i 


APPLICATION    OF   INTEGUATlOX    TO   PLANE   CURVES      339 

7.    Find  the  entire  surface  generated  by  revolving  about  OX  the 
ellipse  3  .r--'  +  4f  =  3a^.  ^^^^^    /3  _^  jn^\,^ 

^■"^  8.   Find  the  entire  surface  generated  by  revolving  about  OY  the 
preceding  ellipse.  ^„^    ^^  _^  3  log  3)^1 

9.    Find  the  surface   generated   by   revolving  about  OX  a  Ujop 
of  the  curve  8  a'7/'-  =  a-x-  —  x*.  a        7r«^ 

^  10.    An  arc,  subtending  an  angle  2  a,  of  a  circle  whose  radius  is  <i. 
revolves  about  its  chord.     Find  the  surface  generated. 

A  n  s.   4  TTfr  ( sin  a  —  a  cos  a) . 

-  11.    The  arc  of  the  catenary  (A rt.  128)  ?/  =  '^(('a-{-  e~«\  from  x  =  a 

to  x=2a,  revolves  about  OY.     Find  the  surface  generated. 

Ans.  (e2  +  2e-'-3e--)7ra2. 

^  12.    The  parametric  equations  of  a  curve  are 

re  =  e*  sin  0,     y  =  e*  cos  6. 

Find  the  surface  generated    by  revolving  the  arc  from  ^  =  0  to 

6=^,  about  OX  Ans.  'tjlljLi^e'" -2). 

13.    Find  the  surface  generated  l)y  revolving  about  01'  the  arc  of 
the  preceding  exami)le.  ^^^^^    2  V2  7r.o  ^-  ^  ^  , 

5 
■^J    14.    The  parametric  equations  of  tlu^  epicycloid,  the  radius  of  tin 
fixed  circle  bein^  «,  and  that  of  the  rolling  circle  -  (Art.  131) 

are   rB  =  '^  cos  <^  -  f  cos3<^,     y  =  ^Sin  </.-^sin3</). 

Find  the  entire  surface  generated  by  revolving  the  curve  about  OX. 

Ans.    '-'n^O'^- 

15.    Find  the  surface  generated  by  revolving  one  arch  of  the  pre- 
ceding curve  about  OF.  -l"^'.    (JTra*. 


340 


INTEGRAL  CALCULUS 


254.  Volume  by  Area  of  Section.  The  volume  of  a  solid  may  be 
found  by  a  single  integration,  when  the  area  of  a  section  can  be  ex- 
pressed in  terms  of  its  per- 
pendicular distance  from 
a  fixed  point. 

Let  us  denote  this  dis- 
tance by  X,  and  .the  area 
of  the  section,  supposed  to 
be  a  function  of  x,  by  X 

The  volume  included 
between  two  sections  sep- 
arated by  the  distance  dx 
will  ultimately  be  Xdx, 
and  we  have  for  the  volume  of  the  solid 

V=  fxdx, 

the  limits  being  the  initial,  and  final,  values  of  x. 


EXAMPLES 

1.    Find  the  volume  of  a  pyramid  or  cone  having  any  base. 

Let  A  be  the  area  of  the  base,  and  h  the  altitude. 

Let  X  denote  the  perpendicular  distance  from  the  vertex  of  a  sec- 
tion parallel  to  the  base.  Calling  the  area  of  this  section  X,  we 
have,  by  solid  geometry, 


X  = 


Axr 


A      h'  /r 

Hence, 


Ah 
3  ' 


2.  Find  the  volume  of  a  right  conoid 
with  circular  base,  the  radius  of  base  being 
a,  and  altitude  h. 

0A  =  BC==2a,     BO=CA  =  h. 

The  section  JiTQ,  perpendicular  to  OA,  is  an  isosceles  triangla 


APPLICATION   OF   INTEGRATION    TO    I'LANK   CLKVES      341 
Let  X  =  OP;  then 

X  =  area  RTQ  =  PTx  PQ  =  h  V2  ax  -  x\ 


Hence, 


ira-h 


This  is  one  half  the  cylinder  of  the  same  base  and  altitude. 


3.    Find  the  volume  of  the  ellipsoid 


Let  us  find  the 
area  of  a  section 
C'B'U  perpendicular 
to  OX,  at  the  dis- 
tance from  the  origin 
OM=x. 

This  section  is  an 
ellipse  whose  semi- 
axes  are  MB'  and 
MC. 

To  find  MB',  let 
2  =  0  in  (1),  aiid  we 
have 


a 


To  find  MC',\ety  =  0\Ti  (1),  and  we  have 


z  =  MC'  =  --^a--x^. 
a 


The  area  of  the  ellipse  (Ex.  2,  p.  321)  is  7r(MB')  (MC). 


Hence, 


X=^  («•-'- or'), 


and  V=  2  r X  dx  =  ^'  ""^'"^  C (a-  -  .r-')  ax  =  i-. 


■abc. 


342  INTEGRAL   CALCULUS 

4.  A  rectangle  moves  from  a  fixed  point,  one  side  varying  as  the 
distance  from  the  point,  and  the  other  as  the  square  of  this  distance. 
At  the  distance  of  2  feet  the  rectangle  becomes  a  square  of  3  feet. 
What  is  the  volume  then  generated  ?  Ans.   4i  cubic  feet. 

J  5.  The  axes  of  two  right  circular  cylinders  having  equal  bases, 
radius  a,  intersect  at  right  angles.  Find  the  volume  common  to 
the  two.     (See  Fig.,  p.  359.)  ^^^     16  a\ 

3 

6.  A  torus  is  generated  by  a  circle,  radius  5,  revolving  about  an 
axis  in  its  plane,  a  being  the  distance  of  the  centre  of  the  circle  from 
the  axis.  Find  the  volume  by  means  of  sections  perpendicular  to 
the  axis.  Ans.    2  Tr'ab-. 

7.  A  football  is  16  inches  long,  and  a  plane  section  containing  a 
seam  of  the  cover  is  an  ellipse  8  inches  broad.  Find  the  volume  of 
the  ball,  assuming  that  the  leather  is  so  stiif  that  every  plane  cross- 
section  is  a  square.  Ans.   341i  cu.  in. 

-r  8.  Given  a  right  cylinder,  altitude  It,  and -radius  of  base  a. 
Through  a  diameter  of  the  upper  base  twf)  i)lanes  are  passed,  touch- 
ing the  lower  base  on  opposite  sides.     Find  the  volume  included 

between  the  planes.  .        f    _^\  y 

ns.    ^--3J«^^. 

V  9.  Two  cylinders  of  equal  altitude  h  have  a  circle  of  radius  a, 
for  their  common  upper  base.  Their  lower  bases  are  tangent  to 
each  other.     Find  the  volume  common  to  the  two  cylinders. 

.         4  a-h 
Ans.    _. 


CHAPTER    XXIX 
SUCCESSIVE   INTEGRATION 

255.  Definite  Double  Integral.  —  A  double  iutegral  is  the  integral 
of  an  integral.  — 

Thus,  X  and  y  being  independent  variables,  the  definite  double 
integral,  ^„  ^, 

1      I    f{x,y)dxdij, 

indicates  the  following  operations:  ^"^ 

Treating   x  as   a   constant,  integrate  fix.  ?/)    with   respect  to  y 

between  the  limits  d  and  c ;  then  integrate  the  result  with  respect 

to  X  between  the  limits  h  and  a.* 
For  example, 

J     J    .-cXft - y)dx dy=^j     .-r(hy  --'A  dx=  \     x' -^ da; 

Notice  that  the  order  of  the  two  integrations  is  indicated  in  the 
given  definite  integral  by  the  order  of  the  difteientials  dxdy,  taken 

/-•2u     /*h 

from  right  to  left,  the  pairs  of  limits  I       I     being  used  in  the  same 
order. 

It  should  be  said,  however,  that  the  order  of  the  integrations  is 
denoted  differently  by  different  writers. 

256.  Variable  Limits.  —  The  limits  of  the  first  integration,  instead 
of  being  constants,  are  often  functions  of  the  variable  of  the  second 
integration. 

*  Using  parentheses,  this  might  be  represented  by  j  " (  i  \f(x,  y)d>j\(lx. 
843 


844  INTEGRAL  CALCULUS 

For  example, 
r  r  xy  dy  clx  =  r(%Y  ydy  =  \  f^S  f  +  2af-  a?y)dy  =  lif  • 

As  another  example, 

_    £S^'^\^  +  y)d^dy=jJ{xy  +  y^^'dx 

When  the  limits  are  all  constants,  as  in  Art.  255,  the  order  of  the 
integrations  may  be  reversed  without  affecting  the  result.     That  is, 

J    x-{h-y)dxdy=\    J     x\h-y)dydx. 

Where  the  definite  integral  has  variable  limits,  the  order  of  integra- 
tions can  be  changed  only  by  new  limits  adapted  to  the  new  order. 

257.    Triple  Integrals.  —  A  similar  notation  is  used  for  three  suc- 
cessive integrations.     Thus 

I      I      I     '^'V"^  f^'^  ^y  dz=  \     I    — —  xr)/-dx  dy 

J(t      Jo     J  a  J^     Jo  2 

2  J^    3  2   ^3       3^        6   ^  '' 


EXAMPLES 

Evaluate  the  following  definite  integrals: 

1-  ££''y^^ - y)'^-'^'^y  =  ^'(« - ^)- 

2.     r  Cr  sin  6  dr  dO  =  ^l^zJl  (cos  y8  -  cos  a). 


SUCCESSIVE  i:\TEGRATION  345 

Jo  Jo    r  <?'•  '^^  =  777-  • 


24 


5.     I      I  r  sm  6  dOdr  =  — -• 

nlOy        


7. 

^0  »/o 

_1, 

2' 

8. 

Jo    Jacose 

9. 

-»)««</,*  =  i. 

<10. 

f"  r  « sin-  <^  di 

H^i^i 

11. 

C"  r^'xdxdy 

X  X  x^  +  f 

=  ^log2. 

12 

J  S  S'^^^ + 2/" + ^)  ^^^  ^^y  '^^  = 

13 

iTX"'- 

'a;^/^)  fZa;  d/y  dz  =  - 

'  >.« 

X'XX"""" 

18 

ahc  ,  , 


15.  rvT 

Jo      Jo  J" 


CHAPTER   XXX 

APPLICATIONS   OF   DOUBLE   INTEGRATION 

258.  Moment  of  Inertia.  If  ?'i,  7*2,  r^,  •••,  r,^  are  the  distances  from 
a  given  liiip.  of  ?*  particles  of  masses  ??i„  m.,,  m..,  •••,  vi„,  the  sum 

^iiiVi^  +  moVo^  +  wio^v  +  •  •  •  +  '>nj\-  =  2^  (j>'i'"i'^ 

is  defined  in  treatises  on  mechanics  as  the  vioment  of  inertia  of  the 
system  about  the  given  line. 

The  moment  of  inertia  of  a  continuous  solid  about  a  given  line  is 
the  sum  of  the  products  obtained  by  multiplying  the  mass  of  each 
infinitesimal  portion  of  the  solid  by  the  square  of  its  distance  from 
the  given  line. 

The  summation  is  then  effected  by  integration,  and  we  have  for 
the  moment  of  inertia  of  a  body  of  mass  M, 


■.jr-,m. 


259.  Moment  of  Inertia  of  a  Plane  Area.  The  moment  of  inertia 
of  a  given  plane  area  about  a  given  point  0  may  be  defined. as  the 
sum  of  the  products  obtained,  by  multiplying  the  area  of  each  infini- 
tesimal portion  by  the  square  of  its  distance  from  0. 

This  may  be  regarded  as  the  moment  of  inertia  of  a  thin  plane 
sheet  of  uniform  thickness  and  density,  about  a  line  through  0  per- 
pendicular to  the  plane,  the  mass  of  a  square  unit  of  the  sheet  being 
taken  as  unity. 

We  shall  illustmte  double  integration  by  findings  the  moment  of 
inertia  of  certain  areas. 


AITLICATIONS   OF   DOUBLE   INTE(niATIO\ 


Wi 


B 

m'  n 

C 

:::[±hH::i:±:b- 

d 

,— [__|.d.._7X~dl-i— 

»                   MM                                AX 

260.  Double  Integration.  Rectangular  Coordinates.  To  liutl  the 
moment  of  inertia  oi'  tlie  rectangle  (JACB  about  O. 

Let    OA^a,  'OB=b. 

Suppose  the  rectangle 
divided  into  rectangular 
elements  by  lines  parallel 
to  the  coordinate  axes.  Let 
X,  y,  which  are  to  be  re- 
garded as  independent  vari- 
ables, be  the  coordinates  of 
any  point  of  intersection  as 
P,  and  X  +  dx,  y^  +  dy  the 
coordinates  of  Q.     Then  the  area  of  the  element  PQ  is  dxdy. 

Moment  of  inertia  of    PQ  =  OP'  •  dx  dy  =  (or  +  _?/-)  dx  dy. 

The  moment  of  inertia  of  the  entire  rectangle  OACB  is  the  sum  of 
all  the  terms  obtained  from  (x-  -}-y-)dx  dy,  by  varying  a;  from  0  to  a, 
and  y  from  0  to  h. 

If  we  suppose  x  to  be  constant,  while  y  varies  from  0  to  b,  we  shall 
have  the  terms  that  constitute  a  vertical  strip  J/iV^VJ/'. 
Hence 

Moment  of  inertia  of    MNN'21'  —  dx  I    (.r  +  ?/-)  dy 

•A) 


=  dx[  x-y  +  -^ 


OX- 


dx. 


Having  thus  found  the  moment  of  a  vertical  strip,  we  may  sum  all 
these  strips  by  supposing  x  in  this  result  to  vary  from  0  to  a.  That 
is,  the  moment  of  inertia  of  OACB, 


■■£ 


,  .  .  'A  ,       n'h  +  ah' 


The   preceding  operations   are  those  represented  by  the    doub 
integral, 

1=  i     I     (:x-  +  y-)dxdy. 


348 


INTEGRAL   CALCULUS 


If  we  first  collect  all  the  elements  in  a  horizontal  strip,  and  then 
sum  these  horizontal  strips,  we  have 


1=  r  r{x'+f)dy 

»/o   »/o 


dx-- 


+  alf 


261.   Variable  Limits.    To  find  the  moment  of  inertia  of  the  right 
triangle  OAC  about  O. 

Let   OA  =  a,  AC^h.     The         y 
equation  of  OC  is 

h 
y  =  -x. 


This  differs  from  .the  pre- 
ceding problem  only'  in  the 
limits  of  the  first  integration. 

In  collecting  the  elements  in  a  vertical  strip  MN,  y  varies  from  0  to 
MN.    But  MN  is  no  longer  a  constant  as  in  Art.  260,  but  varies  with 

OM.  according  to  the  equation  of  0(7,  y  =  -x. 

b  ^ 

Hence  the  limits  of  y  are  0  and  -x. 

a 

In  collecting  all  the  vertical  strips  by  the  second  integration,  x 

varies  from  0  to  a,  as  in  Art.  260. 

Thus  we  have  for  the  moment  of  inertia  of  OAC, 


^=X"X'(^+^'>''^<'^="<f+S' 


By  supposing  the  triangle  composed  of  horizontal  strips  as  HK, 

we  shall  find 

Y 


(ar  +  y"^)  dy  dx 

-J 


"^'T  +  12 


^        APPLICATIONS   OF   DOUHLE   INTEGRATION  349 

262.  Plane  Area  as  a  Double  Integral.  If  in  Art.  260  we  omit  the 
factor  {y? +  ])-),  we  shall  have,  instead  of  the  moment  of  inertia,  the 
area  of  the  given  surface. 

That  is.  Area  =  I    |  '^-f  dij  —  \    \  i.\>j  dx, 

the  limits  being  determined  as  before. 

EXAM  PLES 

1.  Find  the  moment  of  inertia  about  the  origin  of  the  right  tri- 
angle formed  by  the  coordinate  axes  and  the  line  joining  the  points 
(a,  0),  (0,  6).  .  .i^ 

Ans.  pjT   "  {x^  +  f)cUdi,='l!^±J^. 

^    2.    Find  the  moment  of  inertia  about  the   origin  of   the  circle 
ar^  +  2/"  =  «"• 


Ans. 


■  *S'S'""^"('^+fH^"ii=~ 


3.  Find  by  a  double  integration  the  area  between  a  straight  line 
and  a  parabola,  each  of  which  joins  the  origin  and  the  point  {a,  6), 
the  axis  of  X  being  the  axis  of  the  parabola. 

Ans.     r  f     "  dx  dy  =  C  C  dy  dx  =  ^. 

«/u  *yhx  •/o   t/ari^  6 

4.  Find  the  moment  of  inertia  about  the  origin  of  the  preceding 
area.  ,         abfci'  ,  b-\ 

-'"•'•  4(7+5)- 

5.  Find  by  a  double  integration  the  area  included  between  the 
circle  ar  +  ?/^  =  10  ay,  the  line  3  .c  +  ^  =  10  a,  and  the  axis  of  Y. 

lOa-y 

/»«     r'^^ity—y'i  /»10(i     /»      3 

Ans.     I      I  dydx+  I        I  dy  dx 

=  1       I       ,      .    rf-«  (ly  =  -.,  ( '^  +  5  sin     -  )• 


350  INTEGRAL   CALCULUS 

V 

6.    Find  the  moment  of  inertia  about  the  origin  of  the  area  be- 
tween the  ellipse  — f-^  =  l,  and  the  line  -  + '^  =  1. 
a-      I)-  a     b 


^ns.    f^-l^(a'b  +  ab'). 


7.    Find  the  moment  of  inertia  about  the  origin  of  the  area  be- 
tween the  parabola  ay  —  2  (a^  —  x^),   the   circle  x'-  +  f  =  a',  and  the 

axis  of  F.  .         /o2     tt^ 

Ans. 

/ 

•^    8.    Find  by  a  double  integration  the  area  included  between  the 

parabolas  y'  =  S  x,  and  y^  =  12  (60  —  x).  Ans.   900. 


9.    Find  the  moment  of  inertia  of  the  area  included  between  the 

parabola  y"^  —  ^  cix,   x  =  4  a,  and  the  axis  of  X,  about  the  focus  of  the 

parabola.  > 

^  Ans. 


2336^ 
35 


■J 


10.    Find  the  moment  of  inertia  of  the  area  included  between  the 


lines  y  =  2  X,    x -{-  2  y  ■■ 
of  the  first  two  lines. 


5  a,  and  the  axis  of  X,  about  the  intersection 

125  a* 
A71S.    --^^—— 


M' 


^<.R- 


>'Q 


.M 


263.    Double  Integration.     Polar  Coordinates.     To  find  the  area  of 
the  quadrant  of  a  circle  AOB,  whose  radius  is  a. 

In  rectangular  cooi'dinates,  Art. 
260,  the  lines  of  division  consist  of 
two  systems,  for  one  of  which  x  is 
constant,  and  for  the  other,  y  is 
constant. 

So  in  polar  coordinates,  we  have 
one  system  of  straight  lines  through 
the  origin,  for  each  of  which  6  is 
constant,  and  another  system  of 
circles  about  the  origin  as  centre,  for 
each  of  which  r  is  constant. 

Let  r,  6,  which  are  to  be  regarded 
as  independent  variables,  be  the  coordinates  of  any  point  of  intersec- 


X 


N    N'  A 


APPLICATIONS   OF   DOUBLE    INTECiRATION 


351 


tion  as  P,  and  r  +  dr,  0  -\-  (16,  the  coordinates  of  Q.     Then  the  area 

of  PQ  is  ultimately 

PRx  EQ=r(W-dr. 

If  "we  first  integrate,  regarding  6  constant  while  r  varies  from  0 
to  a,  we  collect  all  the  elements  in  any  sector  MOM'. 

The  second  integration  sums  all  the  sectors,  by  varying  6  from  0 

TT 


to 


Hence 


Area  BOA 


S£ 


rd6dr  = 


If  we  reverse  the  order  of  integration,  integrating  first  with  respect 
to  6,  and  afterwards  with  respect  to  r,  we  collect  all  the  elements  in 
a  circular  strip  NLL'X',  and  sum  all  these  strips.     This  is  written 


AXQQ.BOA 


^XT""' 


d6. 


264.  If  the  moment  of  inertia  about  0  is  required,  we  have  for 
the  moment  of  inertia  of  PQ,  r-rdOdr.  Hence,  the  moment  of 
BOA  is 

8 


'-££'■'■"""■=££'' 


drdO- 


265.  Variable  Limits.  To  find  by  a  double  integration  the  area 
of  the  semicircle  OBA\v'\t\\  radius  OC'=a,  the  pole  being  on  the 
circumference. 

The  polar  equation  of  the  circle  is 
r  =  2acos^.  If  we  integrate  first 
with  respect  to  r,  then  with  respect 
to  6,  we  shall  have 


Area  OB  A. 


££ 


rd6dr  = 


Here,  in  collecting  the  elements  in  a  radial  strip  OM,  r  varies 
from  0  to  OM.  P»ut  OM  vhvioa  with  ^,  according  to  the  e(iuation 
of  the  circle  r  =  '2  a  cos  $.     Hence  the  limits  are  0  and  2  a  cos  0. 


B52 


INTEGRAL  CALCULUS 


In  collecting  all  these  radial  strips  for  the  second  integration,  6 
varies  from  0  to  ^  • 

By   supposing  the    area   composed   of   concentric  circular  strips 
al'out  0,  as  LK,  we  find 


Area  OBA 


=17' 


•fe) 


rdrdO  =  ^- 


EXAMPLES 


1.    Find  the  moment  of  inertia  about  the  origin  of  the  area  in- 
'    eluded  between  the  two  circles,  r  —  a  sin  B  and  r  =  h  sin  0,  where  a  >  & 

(Art.    loo).  x.^    /•asine  O 

c/U     «y  i  sin  Q  o2i 

\J    2.    Find  the  moment   of   inertia   about   the    origin   of   the   area 

B 
between  the  parabola  (Art.  139),  r  =  asec--,  its  latus  rectum,  and 

^^-  A71S     '^^«'. 

^''''    "35" 
3.   Find  the  moment  of  inertia  about  its  centre  of  the  area  of  the 


lemniscate  (Art.  143)    r"-'  =  a"  cos 2  B. 


Ans. 


J 


4.    Find  by   double  integration  the   entire  area  of  the   cardioid 
(Art.  141)    r  =  a(l  -  cos  B). 


Ans. 


Sira' 


5.    Find  the  moment  of  inertia  abont  the  origin  of  the  area  of  the 
preceding  cardioid. 


Ans. 


16 


lemniscate  (Art.  14o)  outside  the  circle  2r—a-.       .        ,     , 


AITLICATIOXS   UF    DOUBLE    INTEGKATIUX  o53 

^  6.    Find  the  moment  of  inertia  about  its  centre,  of  the  entire  area 

of  the  four-leaved  rose  (Art.  144)  r  =  a  sin  2  6.  ^3  ira* 

^  Ans. 

IG 

7.    Find   by  a  double    integration   the    area   of   one  loop  of   the 

^{8,.   Find  the  moment  of  inertia   of  the   area  of  the    preceding 
example  about  the  centre  of  the  lemniscate.  .    _    /V^      7r\  a* 

266.  Volumes  and  Surfaces  of  Revolution.  Polar  Coordinates.  If 
in  the  figure  of  Art.  203  we  su})po8e  a  revolution  about  OX,  the 
volume  generated  by  the  infinitesimal  area  PQ  is  the  product  of 
this  area  by  the  circumference  through  which  it  revolves,  that  is, 
2  TT  r  sin  0  -  r  cW  dr. 

Hence  for  the  entire  volume 


V=2,rf  Cr^hie 


dddr, 


the  limits  being  determined  as  in  Art.  263. 
If  the  revolution  is  about  0  Y, 

V=2  7r  C  Cr  COS  Ode  dr. 

The  area  of  the  surface  generated  about  OX  is  (Art.  253) 

^  =  2  TT  fy  ds  =  2irCr  sin  6  p  +  ("^Yl '  dd. 

EXAMPLES 

-    1.   Find  the  volume  generated  by  revolving  the  cardioid  (Art.  141) 

9- =  a  (1  —  cos  ^)  about  OX.     .  8      .,  ^    .      ^,     .        •,     -.      i. 

^  ^  Ans.     - Tra,  twice  the  inscribed  sphere. 

3  ^ 

•^2.    Show  that  the  entire  volume  generated  by  revolving  the  four- 

leaved  rose  (Art.  144)  r=a  sin2  6,  about  OX,  is  =^  of  the  volunioof  tlie 
circumscribed  sphere. 


354  INTEGRAL   CALCULUS 

3.    Find  the  volume  generated  by  revolving  one  loop  of  the  four- 
leaved  rose  r  —  asm26  about  the  axis  of  the  loop. 


Ans.  f8V2-9\^''''' 


105 


N^    4.    Find  the  volume  generated  by  revolving  the  lemniscate  (Art. 
^    143)  ?-  =  a^  cos  2  6  about  0  Y.  ^2  ^3    /t^ 

Ans.      -^. 


5.   Find  the  volume  generated  by  revolving  the  lemniscate  about 


Ans         2 


L<^ '  6.)  Find   area    of   surface   generated   by   revolving   the    cardioid 
^  r = a  (1  —  cos  e)  about  OX.  ^ ._       32  ira^ 


A71S. 


7.    Find   the  moment  of  inertia  of  a   sphere  (radius  a)  about  a 
diameter,  m  being  the  mass  of  a  unit  of  volume.  Ans.      ^  ira^m 

"l5 


f 


CHAPTER    XXXI 

SURFACE,  VOLUME,  AND  MOMENT  OF  INERTIA  OF  ANY  SOLID 

267.    To  find   the  Area  of   Any  Surface,  whose   Equation   is   given 

between  Three  Rectangular  Coordinates,  x,  y,  z. 

Let  this  equation  be 

z=f{x,y). 

Suppose  the  given  surface  to  be  divided  into  elements  by  two 
series  of  planes,  parallel  respectively  to  XZ  and  YZ.  These  planes 
will  also  divide  the  plane  XY  into  elementary  rectangles,  one  of 
which  is  PQ,  the  projection  upon  the  plane  XY  oi  the  corresponding 
element  of  the  surface  P'Q'. 


c 

/ 

/ 

F 
/ 

/ 

> 

y 

Q 

/ 

\ 

M 

N               \ 

P 

y 

yy^               /A          X 

Ik 

_, 

y 

y^ 

L       K 


Let  X,  y,  z  be   the  coordinates  of  P'  and  x-\-dx,  y  +  dy,  z  +  dz, 
of  Q'. 

355 


356  INTEGRAL   CALCULUS 

Since  PQ  is  the  projection  of  PQ',  the  area  of  PQ  is  equal  to  that 
of  P'Q'  mnltiplied  by  the  cosine  of  the  inclination  of  P'Q'  to  the 
plane  XY.  This  angle  is  evidently  that  made  by  the  tangent  plane 
at  P'  with  the  plane  XY.     Denoting  this  angle  by  y, 

Area  PQ   =  Area  P'Q'  •  cos  y, 

Area  P'Q' =  Area  PQ    •  sec  y. 

We  see  from  the  figure  that 

Area  PQ    =  dx  dy. 


Also  from  (8),  Art.  110,    sec  y 


^V-(jMSI 


where  —  and  —  are  partial  derivatives,  taken  from  the  equation  of 
dx         dy 

the  given  surface   z  =f(x,  y). 

Hence  Area  P'Q'  =  [l  +  (^Y  +  (l^)T '^•'^ ^2/- 

If  S  denote  the  required  surface, 

-//t^lJHtJ]'--  •  •  •  (^) 

the  limits  of  the  integration  depending  upon  the  projection,  on  the 
jdane  XY,  of  the  surface  required. 

For  example,  suppose  the  surface  ^IPC  to  be  one  eighth  of  the 
surface  of  a  sphere  whose  equation  is 

x~  +  y-  +  z^  =  a-. 

Here  d^^_x^    §l  =  _l. 

dx  ^       dy  z 


\ 


-(£)■*(!)' 


^      x"  +  y-      a- 


z-  z-      a- 


SURFACE,   A'OLUME,    AND   MOMENT   OF   INERTIA 

Substituting  in  (1),  we  liave 


357 


J  .'  va 


clxdy 


S 


This  is  to  be  integrated  over  the  region  \0B A] the  projection  of 
the  required  surface  on  the  plane  XY. 
The  equation  of  the  boundary  AB  is 

x^  +  y-  =  a-. 

Integrating  first  with  respect  to  ?/,  we  collect  all  the  elements  in 
a  stri-D  M'N'KL,  y  varying  from  zero  to  ML,  that  is,  between  the 
limits  0  and  -y/a^—^- 

Integrating  afterwards  Avith  respect  to  x,  we  sum  all  the  strips,  to 
obtain  the  required  surface  ABC,  x  varying  from  0  to  a. 


Hence 


"££ 


clx  dy 


Va- 


Another  example  is  the  followin 

The  centre  of  a 
sphere,  whose  radius  is 
a,  is  on  the  surface  of  a 
right  circular  cylinder, 
the  radius  of  whose 
base    is    -•      Find   the 

surface  of  the  sphere 
intercepted  by  the 
cylinder. 

Take  for  the  equa- 
tions of  the  sphere  and 
cylinder, 


X-  +  y-  +  ^- 


and 


X-  +  y-  =  ax. 


CPAQ  is  one  fourth 
the  required  surface.     Since  this  surface  is  a  part  of  the  si)here,  the 


358  INTEGRAL   CALCULUS 

partial  derivatives    — ,    —    must  be  taken  from   re- +  w' +  2;^  =  «^ 
ax     dy 

giving,  as  in  the  preceding  example, 

^^  rr      adxdy 
^  ^  Va-  —  T^  —  y^ 

to  be  integrated  over  the  region  OR  A,  the  projection  of  CPAQ  on 
the  plane  XY. 

The  equation  of  the  curve  OR  A  is  x-  -\-y-  =  ax. 

Hence  ^8=  f  f''^'-^^^^^M^=(l-l)a^ 

4         Jo  Jo  ^a'-x'-y-      \2       J 

S  =  (2'n--4:)ar. 

Let  us  now  find  the  surface  of  the  cylinder  intercepted  by  the 
spliere,  one  fourth  of  which  is  CPARO. 

Since  this  is  a  part  of  the  cylinder  x^ -\- y"  ==  ax,  the  partial 
derivatives  in   (1)   must  be  taken  from  this  equation.     But  from 

„       -o  ii    1     dz  dz 

X'  -\-  y-=ax,  we  hnd     —  =00,    —  =  go- 
dx  dy 

The  formula  (1)  is,  then,  inapplicable  in  this  case. 

It  is  also  evident  from  the  figure  that  the  surface  CPARO  cannot 
be  found  from  its  projection  on  the  plane  XY,  since  this  projection 
is  the  curve  OR  A. 

The  difficulty  is  removed  by  projecting  on  the  plane  XZ,  and 
using,  instead  of  (1), 

We  now  find  from  X'  +  y-  =  ax, 

dy  _a  —  2x      dy  _^ 
dx~     2y     '    dz~   ' 

Substituting  in  (2),  and  simplifying, 

4'      J  J  2  ^^c(^c^^^' 


SURFACE,  VOLIMK,   AND   MOMENT   OF   INERTIA        359 


This  must  be  intej^rated  over  the  region  C'P'AO,   CP'A  being  the 
projection  on  XZ  of  CPA. 

To  find  the  equation  of  CP'A,  we  eliminate  y  from 


giving, 
Hence 


xr  +  y-  +  z-  =  a-   and   cii^  +  y^  =  ax^ 
z-  =  a-  —  ax. 


i-=iX"X 


Vax  —  x^ 


=  a-,  S  =  4  a^. 


EXAMPLES 

1.    The  axes  of  two  equal  right  circular  cylinders,  a  being  the 
radius    of    base,    intersect    at    right 
angles  ;  find  the  surface  of  one  inter- 
cepted by  the  other. 

Take    for    the    equations    of    the 
cylinders, 

or  +Z-  —  d\   and   ^  -\-i^—  «-. 

Am.     8  a\ 

\   2.  Find  the  area  of  the  part  of  the 
plane 

a     b      c 


in  the  first  octant,  inter- 
cepted by  the  coordinate 
planes.  ■ 


3.  Find  the  area  of  the 
surface  of  the  cylinder 
oy^  +  t/2  =  a-,  included  be- 
tween the  plane  z=nix  and 
the  plane  XF.     ^4,,,,  4^«2. 


360 


INTEGRAL   CALCULUS 


7    4.    Find  the  area  of  the  surface  of  the  paraboloid  of  revohition 
_?/-  +  z^  =  4  ax,   intercepted  by  the 
parabolic  cylinder    y^ 
the  plane    x  =  oa. 


Am 


ax,    and 

56  Tra^ 


5.  In  the  preceding  example, 
find  the  area  of  the  surface  of 
the  cylinder  intercepted  by  the 
paraboloid  of  revolution  and  the 
given  plane. 

Ans.     (13^/13-1) -73. 


J.    6.    Find  the  area  of  that  part  of  the  surface 

z^  -f-  {x  cos  a  +  y  sin  of  =  a^ 

which  is  situated  in  the  first  octant. 

The  surface  is  a 
right  circular  cyl- 
inder, whose  axis 
is  the  line  z  =  0, 
X  cos  a+y  sin  a=0, 
and  radius  of  base  a. 
a- 


Ans. 


sm  a  cos  a 


7.  A  diameter  of 
a  sphere  whose 
radius  is  a  is  the 
axis  of  a  right 
prism  with  a  square 
base,  2  h  being  the 
side  of  the  square, 
the  prism. 


Find  the  surface  of  the  sphere  intercepted  by 


Ans. 


<ah 


b  sin" 


b 


.-1     b' 


SIJUFACE,    VOLl'MH,    AND    MOMHX'l'   OF    IXKRTTA       361 


268.  To  find  the  Volume  of  Any  Solid  bounded  by  a  Surface,  whose 
Equation  is  given  between  Three  Rectangular  Coordinates,  x,  y,  z. 

The  solid  may  be  supposcil  to  be  divideil,  by  planes  parallel  to  the 
coordinate  planes,  into  elementary  rectangular  paralleloj)ipeds.  The 
volume  of. one  of  these  parallelopipeds  is  dxdi/dz,  and  the  volume  of 
the  entire  solid  is  r  r  r 

the  limits  of  the  integration  depending  upon   the  equation  of  the 
bounding  surface. 

For  example,  let  us  find  the  volume  of  one  eighth  of  the  ellipsoid 
whose  equation  is  ^ 


X 

PQ  represents  one  of  the  elementary  parallelopipeds  whose  volume 
is  dx  dy  dz. 

If  we  integrate  with  respect  to  z,  we  collect  all  the  elements  in  the 
column  MN',   z  varying  from  zero  to  MM';  that  is, 


from   0   to   z  =  cJi_^_^t- 
V        a-      Ir 


862  INTEGRAL   CALCULUS 

Integrating  next  with  respect  to  y,  we  collect  all  the  columns  in 
the  slice  KLN'H,  -y  varying  from  zero  to  KL ;  that  is, 

from   '^   -~    -       '     '-      ^ 


0    to   y  =  h\\l--^. 
^         d- 


This  value  of  y  is  taken  from  the  equation  of  the  curve  ALB, 
which  is 

"^  +  1  =  1. 

Finally,  we  integrate  with  respect  to  x,  to  collect  all  the  slices  in 
the  entire  solid  ABC.  Here  x  varies  from  zero  to  OA ;  that  is,  from 
0  to  a. 

The  y  and  x  integrations  are  said  to  be  over  tlAe  region  AOB. 


have    r=  rf^'^Z  C'''^?---d.dydz  =  ^L^. 


For  the  entire  ellipsoid     V=  il^. 


EXAMPLES 

r^.    Find  tlie  volnme  of  one  of  the  wedges  cut  from  the  cylinder 
X-  +  y^—  a^  by  the  plane  z  =  x  tan  a  and  the  plane  XY.    (See  Figure, 

Ex.  3,  Art.  267.)  ^    ^  , ^  ,  „    „  ^ 

dx  dy  dz  = . 


^'-•^XX    X 


N^    2.    Find  the  volume  of  the  tetrahedron  bounded  by  the  coordinate 
planes  and  by  the  plane 

^  +  ^  +  ?  =  l.  Ans.^. 

a      b     c  6 

3.  Find  the  volume  included  between  the  paraboloid  of  revolution 
y- +  z^  —  4 ax,  the  parabolic  cylinder  ?/-  =  aa;,  and  the  plane  x=3a. 
(See  Figure,  Ex.  4,  Art.  267.)  Ans.  (6  tt  +  9  V3)rfl 


SURFACE,   VOLUME,   AND  MOMENT   OF   INERTIA       363 

*^  4.   Find  the  volume  contained  between  the  paraboloid  of  revolution 
ar  +  //'-  =  az,  the  cylinder  x-  +  y-  =  2  ax,  and  the  plane  XY. 

5.   Find  the  volume  of  the  cylinder  x-  +  y^  —  ax,  intercepted  by  the 
paraboloid  of  revolution  y-  -\-z-  =  2  ax. 


(f-i>'- 


^  6.  The  centre  of  a  sphere  (radius  a)  is  on  the  surface  of  a  right 
circular  cylinder,  the  radius  of  whose  base  is  -.  Find  the  volume  of 
the  part  of  the  cylinder  intercepted  by  the  sphere.    (See  second  Figure, 


3V        3 

7.   Find  the  volume  in  the  first  octant,  bounded  by  the  surface 

,aj 


h         \cj  90' 


^  8.   Find  the  entire  volume  within  the  surface 

x3  ^yi  -^z^  —  a^.  Ans.  -^    . 

So 

269.  Moment  of  Inertia  of  Any  Solid.  This  may  be  expressed  by  a 
triple  integral. 

Thus,  the  moment  of  inertia  about  OX,  m  being  the  mass  of  a  unit 
of  volume,  is 

/  =  m  f  f  f  (.'/"  +  2")  dx  dy  dz, 

with  similar  formulae  for  the  moments  of  inertia  about  the  axes 
OY,  OZ. 

EXAMPLES 

1.  Find  the  moment  of  inertia  about  OX  of  the  rectangular  paral- 
lelopiped  bounded  by  the  planes  x  =  a,  y  =  h,  z  —  c,  and  the  co- 
ordinate planes.  ■  ^luj.  (b-  I  r)^'"^^. 

3 


364  INTEGRAL   CALCULUS 

J 

2.  Find   tlie   moment   of   inertia  about    OZ   of  the  tetrahedron 
bounded  by  the  plane 

a     h      e 
and  by  the  coordinate  planes.  Ans.  (a-  +  ^')~^~  • 

3.  Find  the  moment  of  inertia  about  OX  of  the  portion  of  the 
cylinder  a;^  +  3/^  =  a^  included  between  the  planes  z  =  li  and  z=  —  7i. 

.  nj  fa-  ,  2  h- 

^  4.    Find  the  moment  of  inertia  of  the  preceding  cylinder  about  OZ. 

Alts.  TT/ncf'/i. 

5.    Find  the  moment  of  inertia  of  a  sphere  (radius  «)  about  a 
diameter.  .   ^    Svina^ 


15 
v/   6.    Find  the  moment  of  inertia  about  OZ  of  the  ellipsoid 

a-     b^     c^  15 


CHAPTER   XXXII 

CENTRE    OF   GRAVITY.     PRESSURE    OF   FLUIDS 
FORCE    OF   ATTRACTION 

CENTJ^E   OF   GRAVITY 

270.  Definition.  The  centre  of  gravity  of  a  body  is  a  point  so 
situated  that  the  force  of  gravity  acting  on  the  body  produces  no 
tendency  to  rotate  about  an  axis  passing  througli  the  point. 


271.  Coordinates  of  Centre  of  Gravity.  To  find  the  centre  of 
gravity,  C,  of  any  body,  take  P  as  any  infiuitesiuial  part  of  the 
given  body,  PQ  the  line 
of  direction  of  gravity, 
and  MN  any  horizontal 
axis  passing  through  C. 
Let  BD  be  the  common 
perpendicular  between 
MN  and  PQ.  Take  the 
axis  of  X  parallel  to  BD 
and  represent  by  x  and 
X,  OL  and  OL',  the  x 
coordinates  of  P  and  C  re- 
spectively. Then  the  dis- 
tance BD  =  L'L  =  x  —  X. 

The  force  exerted  by  gravity  on  P  is  proportional  to  and  there- 
fore may  be  measured  by  its  mass.  Denoting  its  mass  by  dm,  the 
moment  of  this  force  about  J/X  would  be  (x  —  x)dm;  and  if  dm  ij 
an  infinitesimal  in  one,  two,  or  three  dimensions,  the  tendency  of 

the  whole  body  to  rotate  about  MX  is  equal  to   |  (x  —x)dm. 

365 


S6G  INTEGRAL   CALCULUS 

Since  this  must  equal  zero, 

(.c  —  x)dm  —  0, 

I  xdm 


/< 


(1) 


dm. 


Similar  formulge  may  be  derived  for  y  and  z. 

Note.  — The  mass  of  a  unit's  volume  is  called  density.  If  we 
represent  the  density  by  p,  the  differential  mass  or  dm  is  equal  to  p 
multiplied  by  the  differential  of  the  arc,  area,  or  volume, 

Ex.  1.     Find  the  centre  of  gravity  of  a  quarter  of  the  arc  of  a  circle. 

Let  the  equation  of  the  cii'cle  be  x^  +  ?/-=  al 

Here  dm  =  pds. 

Substituting  in  (1),  Art.  271,  we  have 


I  x'rfs      at    x(a- —  ar)' 
fds  I  .a 


^-dx 


From  the  symmetry  of  the  figure  y  =  ' 


Ex.  2.     Find  the  centre  of  gravity  of  the  surface  bounded  by  a 
parabola,  its  axis,  and  one  of  its  ordinates. 

Let  the  equation  of  the  parabola  be  2/^  =  4paJ,  B  being  (9j),  6p). 
Here  dm  =  pdxdy,  and  substituting  in  formula  (1),  Art.  271. 


J      I     ^""x  dx  dy 
^     0     »/o • 

I      i    ""dxdy 

V4^    I       X^dx        r,r- 


x^dx 


} 

CENTRE   OF   GRAVITY.     PRESSURE   OF   FLUIDS        367 

Similarly, 


a;  dx 


I         dxdy       V4j9  I     x 
Jo  Jq 


idx 


Ex.  3.    Find  the  centre  of  gravity  of  a  circular  disk  of  radius 
a,  whose  density  varies  directly  as  the  distance  from  the  centre,  and 
from  which  a  circle  described 
upon  a  radius  as  a  diameter 
has  been  cut. 

Let  the  equation  of  the 
large  circle  be  7-  =  a-,  and 
the  equation  of  the  small 
circle  be  r  =  —  a  cos  9. 

The  disk  is  symmetrical 
with  respect  to  OX,  hence 

Here 

dm  =  pr  dO  dr  =  k/--  dO  dr, 

(\ip  =  Kr). 
Also  X  =  031=  r  cos  6. 


C^  f"  r  COS  0  de  dr  +   C  C^  ?'^ 

*/U     «yO  Jtt     J -a  rose 

Therefore  ^  =  '^      


cos  0  dO  dr 


n  rr-dedr+  r  f      rdddr 

»/0      Jo  ,  t/'J-      J -a  cone 

J  r  p  cos  6  dd  +   C  (cos  e  -  cos'^)  dd  1 

5:=: 'i — 

'1[J\W+  f^  ( 1  +  cos^^)  dd\ 


6a 

5(37r-2j 


=  o.ir,ir,  a. 


3G8 


INTEGRAL   CALCULUS 


Ex.  4.     Find  the  centre  of  gravity  of  a  cone  of   revolution,  the 
radius  of  the  base  being  2  and  the  altitude  6. 
The  equation  of  OB  is  y  =  J-  x. 
Here  dm  =  p-n-y-  dx,  and  substituting  in  (1),  Art.  271, 


dx 


y-  dx 


The  cone  is  symmetrical  with 
respect  to  OX,  hence  y  =  0. 

Note. — On  comparing  the 
formulae  for  the  centre  of  grav- 
ity of  arc,  area,  and  volume, 


Y 

x_ 

X'V\ 

3 

\il\ 

; 

X 

fxds  CxclT'  fxdV 

■  ^ -,     X='^ ,     x=^ , 

fas  faA  fcv 


we  notice  that,  in  each  case,  the  element  of  the  numerator  integral 
is  X  times  the  element  of  the  denominator  integral. 

[^5.    Find    the   centre  of   gravity    of   the  arc  of    the    hypocycloid 

9  2  2  __  _  ^> 

(Art.  1.S2)  x'i  4-?/3=a^  in  the  first  quadrant.  Ans.    x  =  y  =  ^a. 


j^6.  Find  the  centre  of  gravity  of  the  arc  of  the  cycloid 


x  =  a  vers     --^  —  V2  ay  —  y'i 


Aiis.   x  =  Tra,  2/  =  o  '^• 

o 

-f  7.  Find  the  centre  of  gravity  of  a  straight  rod  of  length  a,  the 
density  of  which  varies  as  the  third  power  of  the  distance  of  each 
point  from  one  end.  ^^^^^    x  =  -a. 

5 

^^8.  Find  the  centre  of  gravity  of  the  surface  of  a  hemisphere 
when  the  density  at  each  point  of  the  surface  varies  as  its  perpen- 
dicular distance  from  the  base  of  the  hemisphere. 


Ans.   X  = 


CEX  PRE    OF   CKAVITY.      rilKSSLKK    OF    FI.IIDS         :]{)[} 


(T7  9.'  Fiiid.  the  centre  of  gravity  of  a  seraiellipso. 

o 


.1...     '" 


j^  10.    Find  the  centro  of  gravity  of  the  area  between  the   cissoid 
V^  r. 

y^j= — '- and  its  asyu^ptote.  Ans.   x  =  '-  a. 

%    2  a  —  X  o 

11.  Find  the  centre  of  gravity  of  the  area  bounded  by  the  paial)- 
ola  y-  =  8  X,   the  line  //  +  .i-  —  G  =  0,    and  the  axis  of  X. 

An,s.    x=^:4&ry=iAr- 

/->  12.    Find  the  centre  of  gravity  of  one  loop  of  the  curve  r  =  a  sin  2  $. 

^C:                                                                      ,                128  a    _      128  a 
>  Ans.    X  = :  y  = • 

^  105  TT       ^  105  TT 

13.  Find  the  centre  of  gravity  of  the  upper  half  of  the  cardioid 
'•  =  a{l-cose).  ^^^^    -^_5 

0 

_  Ifi  a  ;-(-_ . 

— -^TT 

14.  Find  the  centre  of  gravity  of  one  loop  of   the   lenniiscate 

,•2  ^ tr  cos  2  9.  .        -     7r^'  2 ,,       --  ^ 

Ans.   X  =  — —  a  =  .oo  a. 
8 

3 

15.  Find  the  centre  of  gravity  of  a  hemisphere.         Aus.  .r  =  -  ((. 

'^  16.    Find  the  centre  of  gravity  of  a  heniispheroid.      Ans.    x  =  -o. 

17.  A  right  cone  of  height  h  is  seooited  out  of  a  right  cylinder  of 
the  same  height  and  base.  Find  the  distance  of  the  centre  of  gravity 
of  the  remainder  from  the  vertex.  ^.j^^^^  "^ ;, 


272.    Theorems  of  Pappus. 

TliooKun  T.  If  a  i)lane  area  be  revolved  about  an  axis  in  its  plane 
and  not  crossing  the  area,  the  volume  of  the  soliil  generated  is  equal 
to  the  product  of  the  area  and  the  length  of  the  path  described  by 
the  centre  of  gravity  of  the  area. 


370  INTEGRAL  CALCULUS 

Theorem  11.  If  the  arc  of  a  curve  be  revolved  about  an  axis  in 
its  plane  and  not  crossing  the  arc,  the  area  of  the  surfaoe  generated  is 
equal  to  the  product  of  the  length  of  the  arc  and  the  path  described 
by  the  centre  of  gravity  of  the  arc. 

273.  Proof  of  the  Theorems.  Let  the  area  be  in  the  plane  XY 
and  let  it  revolve  about  the  axis  of  X.  Then  by  (1),  Art.  271,  we 
have 


2/  = 


\   i  y  dx  dy 
I   I  dxdy 


2ny  ^^dxdy^jJ2^ydxdy (1) 


Then 


But  the  right-hand  member  of  equation  (1)  is  the  volume  described 
by  revolving  the  area  through  the  angle  2  it,  2  Try  is  the  length  of  the 

path  described  by  the  centre  of  gravity,  and    |    |  dx  dy  is  the  plane 
area. 

The  first  theorem  is  thus  seen  to  be  true,  and  the  second  can  be 
proved  true  in  a  similar  manner. 

EXAMPLES 

1.  Find  the  volume  and  surface  generated  by  revolving  a  rec- 
tangle with  dimensions  a  and  b  about  an  axis  c  units  from  the  centre 
of  the  rectangle.  ^,^^.    2  .rabc,  and  4  ,r(a  +  b)c. 

2.  Find  the  volume  and  surface  generated  by  revolving  an  equi- 
lateral triangle  each  side  a  units  in  length,  about  an  axis  c  units 
from  the  centre  of  the  triangle.  •  ^    /o 

Ans.  "I^L^XE  and  6  Trac. 


CENTRE   OF   (illAVII  Y 


'RESSUllE   OF   FLUIDS 


371 


3.  Find  the  volume  and  surface  generated  by  revolving  a  circle 
of  radius  a  about  an  axis  b  units  from  the  centre  of  the  circle. 

Ans.   2  TT^a-b  and  4  ir'ab. 

4.  Find  the  volume  generated  by  revolving  an  ellipse,  semiaxes 
a  and  b,  about  an  axis  c  uuits  from  the  centre  of  the  ellipse. 

Ans.    2  ir'-'uhr. 


PRESSURE  OF   LIQUIDS 

274.  The  pressure  of  a  li(]uid  on  any  given  horizontal  surface  is 
eqiuil  to  the  weight  of  a  cohunn  of  the  liquid  whose  base  is  the 
given  surface  and  whose  height  is  equal  to  the  distance  of  this  sur- 
face below  the  surface  of  the  liquid. 

The  pressure  on  any  vertical  surface  varies  as  the  depth,  and  the 
method  of  determining  it  is  illustrated  by  the  following  exauiples. 

Ex.  1.  Suppose  it  is  required  to  find  the  pressure  on  the 
rectangidar  board  OABC,  the  edge  OC  being  at  the  surface  of  the 
water. 

Let  BC=a,  and  AB  =  b. 

Suppose  the  rectangle  divided 
into  horizontal  strips  one  of  which 
is  HK. 

Let  OH=x,  then  the  width  of 
the  strip  is  dx.  H 

If   the    pressure    on    this    strip 
were  uniform   thrQugliout  and  the         A 
same  as  it 
the  pre 
icbx- 
a  cut 


p  of  the  strip, 

'strip  woidd  be 

is  the  weight  of 

Qf  the    water.     And 

ressure  on  the  board  is  evidently  the  integral  of  this 


0 

2 

Y 

X 

a 

K 

B 

b 

That  is.       Entire  pressure 


=1 


:bx  dx : 


g-bw 
2 


INTEGRAL   CALCULUS 


Ex.  2.  Find  the  pressure  on  that  pnrt  of  the  board  in  Exam- 
ple 1,  which  is  below  the  diagonal. 

In  this  case  the  area  of  HKis  y  dx,  and  the  entire  pressure  on  the 
triangular  board  is 


X 


But 


v:yx  dx. 
b  . 


hence  entire  pressure 

bw  C  ■>  7        bn^i 

=  —  I    x-dx=  — 

a  Jo  o 


nixr 

IT'        x 

Note.  — Call  the  weight  of  a  culiie  foot  of  water,  02  ll)s. 

Ex.  3:    One  face  of  a  ])0x  immersed  in   water  is  in  the  form  of 
a  square,  the  diagonals  being  8  feet  in  length.     The  centre  of  the 
square  is  6  feet  below  the  surface  of  the  water,  and  one  diagonal  is 
vertical.    Find  the  pressure 
on  the  sq\iare  face. 

Let  S  W  be  the  surface 
of  the  water.  Taking  the 
axes  as  in  the  figure,  the 
equations  of  AB  and  BC 
are  y=4:-\-x,  and  ?/=4  — cc, 
respectively. 

Then,  if  P  represents  the 
Piitire  pressure  on  the 
ijiiard. 


>==2wC   r      {G  +  x)dydx 

Jo    Jy-i 

=  192  «<  =  11904  lbs. 


Ex.  4.  Find  the  pressure  on  a  sphere  6  feet  in  diameter,  im- 
mersed in  water,  the  centre  of  the  sphere  being  10  feet  below  the 
surface  of  the  water. 


CKNTllE    OF   CillAVITV.      I'llESSllii:    Ol'    FLIIDS        373 

Let  /STFbe  the  surface  of     S i W 

the  water. 

Take  the  axes  as  in  tlie 
figure,  and  let  tKe  elemen- 
tary surfae^'^be  a  zone. 
The  'gcmi  of  a  zone  at  a 
distance  x  from  the  cen- 
tre of  the  sphere  is  2  iry  ds. 
The  pressure  on  the  zone 
is  2  TTioy  (10  -f-  x)ds. 

Then,  if  P  represents 
the  entire  pressure  on  the 
sphere, 

P=2-,rw 


^jll{10  +  x)(h. 


But 


y  =  Vy  —  x',  and  ds  =  '-  dx. 


1  lence 


P  =  Gtt  If  C    (10  +  x)dx, 
=  3G0  TTir  =  223207r  lbs. 


5.  A  rectangular  flood  gate  whose  iipper  edge  is  in  the  surface  of 
the  water,  is  divided  into  three  i)arts  by  two  lines  from  the  middle 
of  lower  edge  to  the  extremities  of  upper  edge  Show  tliat  the  parts 
sustain  equal  pressures. 


upper  edga^in  ■ 
doubl#1lh|Jrei 


Ahs.   3  ft. 


6.  A  rectangular  flood  gate  10  feet  broad  and  0  feet  deep  has  its 
upper  edga^in  the  surface  of  the  water.     How  far  must  it  be  sunk  to 

'ssure ' 

7.  A  board  in  the  form  of  a  parabolic  segment  by  a  chord  j.erpen- 
dicular  to  the  axis  is  immersed  in  water.  Tht;  vertex  is  at  the  sur- 
face and  the  axis  vertical.  It  is  20  feet  deep  and  12  feet  broad. 
Find  the  pressure  in  tons.  -1"^-    •"'''.>-">2. 


374  INTEGRAL   CALCULUS 

8.  How  far  must  the  board  iu  Ex.   7  be  sunk  to  double  tl 
pressure  ?  Ans.    12  i 

9.  Suppose  the  position  of  the  parabolic  board  in  Ex.  7  reverse 
the  chord  being  in  the  surface ;  what  is  the  pressure  ? 

Ans.   39.68  ton 

10.  How  far  must  the  board  in  Ex.  9  be  sunk  to  double  tl 
pressure  ?  Ans.    8  f 

11.  A  trough  2  feet  deep  and  2  feet  broad  at  the  top  has  sera 
elliptical  ends.     If  it  is  full  of  water,  find  the  pressure  on  one  en( 

Ans.    1651  lb 

12.  One  end  of  an  unfinished  water  main  2  feet  in  diameter 
closed  by  a  temporary  bidkhead  and  the  water  is  let  in  from  tl 
reservoir.     Find  the  pressure  on  the  bulkhead  if  its  centre  is  30  fei 
below  the  surface  of  the  water  in  the  reservoir.  Ans.    18607r  lb 

13.  A  water  tank  is  iu  the  form  of  a  hemisphere  24  feet  in  diam 
ter  surmounted  by  a  cylinder  of  the  same  diameter  and  10  feet  hig] 
Find  the  total  pressure  on  the  surface  of  the  tank  when  the  tank 
filled  to  within  2  feet  of  the  top.  a         744  -n-  .^^ 

5 

14.  A  cylindrical  vessel,  whose  depth  is  12  inches  and  base 
circle  of  20  inches  diameter,  is  filled  with  equal  parts  of  water  an 
oil.  Assuming  the  oil  to  be  half  as  heavy  as  the  water,  show  th: 
the  pressure  on  the  base  equals  the  lateral  pressure. 

275.  Centre  of  Pressure.  Since  the  pressure  of  a  liquid  on  a  verij 
cal  surface  varies  as  the  depth,  there  exists  a  horizontal  line  aboi] 
which  the  statical  moment  of  the  entire  pressure  on  the  surface  | 
zero.  Such  a  line  passes  tlirough  the  centre  of  pressure  and  tl 
abscissa  of  this  point  may  be  found  by  the  method  used  in  the  foUoN 
ing  example. 

Ex.  1.  Find  the  abscissa  of  the  centre  of  liquid  pressure  i 
a  vertical  surface  bounded  by  the  curve  y  =  f(x),  the  axis  of  X  ai 
the  two  ordinates  ?/o  and  y^.     Given  that  the  origin  is  at  a  distan 


CENTRE   OF  GRAVITY.     PRESSURE   OF   FLUIDS 


r 


Or 


icy  (7i+x)  (x—x)  dx=0. 

I    '  xy  (h  +  x)  dx 

x='^^ 

I    '  y  (h  +  x)  dx 


-W 


ft  below  the  surface  of  the  liquid,  the  axis  of  X  vertical,  and  the 

weight  of  a  (uibic  unit  of  liquul  is  re. 

Let  FoPiRQ  be  the  surface  bounded  by  the  curve  y=f(x),  the 

axis  of  X,  and  the  two  ordinates  ?/o  =  QPo  and  y^  =  JiPi.     Divide 

the  surface  into  horizontal  strips  of  width  dx,  one  of  which  is  IIK. 

Let  OH—x.     Let  MN  pass  through  the  centre  of  liquid  pressure, 

and  OM  =  X. 

Then  the  pressure  on  the 

S 

strip  HK  is  ivy  (h+  x)  dx, 

and  the  moment  of  this 
pressure  about  3/jV  is 
tvy  (h  -f-  x)(x  —x)  dx. 
Therefore,  the  moment 
of  the  entire  pressure  is 
the  integral  of  this  ex- 
pression between  the  ab- 
scissas of  Po  and  Pi,  that 
is,  between  Xq  and  a^. 
But  this  must  equal  zero, 
therefore 


Y 

1        1       p^ 

N 

g 

1         \x          \ 

M 

Pi 

K 
X 

2.  Find  the  centre  of  pressure  of  the  water  on  the  parabolic 
board  given  in  Ex.  7,  Art.  274.  ^l^'.s.    14^  it.  below  vertex. 

3.  Find  the  centre  of  pressure  of  the  water  on  the  bulkhead  given 
in  Ex.  12,  Art.  274.  Ans.  j\  inch  below  centre  of  bulkhead. 

4.  A  rectangular  flood  gate  a  feet  deep  and  h  feet  broad,  with  its 
ipper  edge  at  the  surface,  is  to  be  braced  along  a  horizontal  line. 
How  far  down  must  the  brace  be  put  that  the  gate  may  not  tend  to 
-urn  about  it?  Ans.  I  a  ft. 


376  INTEGRAL   CALCULUS 

5.  One  end  of  a  cylindrical  aqueduct  6  feet  in  diameter  which 
half  full  of  water  is  closed  by  a  water-tight  bulkhead  held  in  plf 
by  a  brace.  How  far  below  the  centre  of  the  bulkhead  should  t 
brace  be  put  ?     What  pressure  must  it  be  able  to  withstand  ? 

Ans.  x  =  j%7rlt;  P=  1116  ' 

6.  A  water  pipe  passes  through  a  masonry  dam,  enters  a  reserve 
and  is  closed  by  a  cast-iron  circular  valve  which  is  hinged  at  t 
top.  The  diameter  of  the  valve  is  3  feet,  and  the  depth  of  its  cen 
below  the  water  level  in  the  reservoir  is  12  feet.  Find  the  press; 
on  the  valve,  and  the  distance  of  the  centre  of  pressure  below  1 
hinge.  Ans.  P=  1674  tt  lbs.  and  f  | 

7.  Water  is  flowing  along  a  ditch  of  rectangular  section  4  f^ 
deep  and  1  foot  wide.  The  water  is  stopped  by  a  board  fitting  \ 
ditch  and  held  vertical  by  two  bars  crossing  the  ditch  horizontal 
one  at  the  bottom  and  the  other  one  foot  from  the  bottom  of  1 
ditch.  How  high  must  the  water  rise  to  force  a  passage  by  ups 
ting  the  board  ?  Ans.    To  within  1  ft.  of  top  of  dit( 

ATTRACTIOX   AT   A  POINT 

276.  A  particle  of  mass  m  is  situated  at  a  perpendicular  distai 
c  from  one  end  of  a  thin,  straight,  homogeneous  wire  of  mass  3/ a 
length  I.  Eequired  to  find  the  attraction  on  the  particle  due  to  i 
wire. 

Let  0  be  the  particle  and  AB  the  wire.  Let  X  and  Y  be  i 
components  of  the  attraction  along  the  axes  of  X  and  Frespective 

Divide  AB  into  elements  of  length  dy 
and  let  PQ  be  one  of  these  elements. 

The  mass  of  PQis    —dy,  since—  is 

the  mass  of  a  unit's  length. 

If  the  mass  of  PQ  were  concen- 
trated at  P,  the  attraction  at  0  due 
to  PQ  is,  according  to  Newton's  Law 

of  Attraction,  nr  i 

'  Kinmdy 

Ko'  +  yi 


CENTRE   OF   GRAVITY.     PRESSURE  OF   FLUIDS         o77 
nd  the  components  along  OX  and  0  Y  are 


respectively, 
ft 


-^^^^1^^^  cose,  and  ^^^^^^1^  sin 


Substituting  for  cos  9  and  sin  6  their  values  we  have 
I  Mr  r^      (hi       _     Kin  M    _Km. 


^^_^.  .,„        ^     KmM     ^-mE^ina. 


Mr  r^       (hi        _     KinM     ^ 

KmM ,^      „  „    \ 

= (1  —  cos  «), 

cl 
where  the  angle  AOB=  a. 

Denoting  by  R  the  total  attraction  of  the  wire  on  the  particle, 
J,         , —     KinM    , 

■R=Vx-+  y-=— ^V2(i-cos«) 


2KmM  .    1 

sni    «. 


The  line  of  attraction  evidently  makes  with  OA  an  angle  whose 
tangent  is 

Y     1  —  cos  a      .      1 

—  = : =  tan-a. 

A         sm  u  2 

The  resultant  attraction,  therefore,  bisects  the  angle  «. 

Note.  —  Tf  we  take  as  our  unit  of  force  the  force  of  attraction  be- 
tween two  unit  masses  concentrated  at  j)oints  Avhich  are  at  unit's 
distance  apart,  k  becomes  unity. 


378  INTEGRAL   CALCULUS  ?  ^ 


EXAMPLES 


1.   Find   the  attraction  perpendicular  to  the  wire  in   preceding 

example  when  the  particle  is  at  a  distance  -  above  0 

o 


c     LV9C-  +  4Z2      -Vdc'  +  PJ      k 


2.  Find  the  attraction  of  a  thin,  straight,  homogeneous  wire  of 
length  I  and  mass  ilf  upon  a  particle  or  mass  m  which  is  situated  at 
a  distance  c  from  one  end  of  the  wire  and  in  its  line  of  direction. 

KmM 


Ans. 


c{c  +  l) 


3.  Find  the  attraction  of  a  homogeneous  circular  disk  of  radius  a 
upon  a  particle  of  mass  m  in  its  axis  and  at  a  distance  c  from  the 
disk. 

Ans.  2  KTrmpl  1  —  — z==-  \,  where  p  is  the  mass  of  the  disk  corre- 
L        -Vc'  +  a^J 
spending  to  units'  surface. 


4.  Find  the  attraction  due  to  a  homogeneous  right  circular  cylinder 
of  length  2  /  and  radius  a  upon  a  mass  ??i  in  the  axis  produced  of  the 
cylinder  and  distant  c  from  one  end. 


Ans.  2  TTKiiip  [2 1  +  Vcr  +  c^  -  Va^  +  (c  +  2  /)  -] . 


CHAPTER   XXXIII 

INTEGRALS   FOR  REFERENCE 

277.   We  give  for  reference  a  list  of  some  of  the  integrals  of  the 
preceding  chapters. 

1.    Cx^dx  =  ^^- 

J  n  +  1 

2.  r^=iogx. 

J    X 

3.     r-^„  =  ltan-^. 

J  .f-  -f  w      a  a 

4C    dx  1    ,      X—  a 

J  X-  —  a-      'J a        X -{-a 

EXPONENTIAL  INTEGRALS 

5.  Ca-dx  =  -^- 
J  log  a 

6.  Ce^  dx  =  e'. 

TRIGONOMETRIC    INTEGRALS 

7.  I  sin  X  dx  =  —  cos  x. 


8.     I  cosxdx=:  sin; 


370 


380 


INTEGRAL   CALCULUS 
9.      I  tan  X  (Ix  =  ]  og  sec  x, 

10.  I  cot  X  dx  —  log  sin  x. 

11.  I  sec  x"  r^a;  =  log  (^  sec  a;  +  tan  x) 


:tan[ • 


V4      li 


12. 


j  cosec  X  (Zx  =  log  (cosec  x  —  cot  x) 


=  log  tan  ^. 

13.  I  sec- cc  dx  =  tan  a;. 

14.  I  cosec^  .i;  dx  =  —  cot  x. 

15.  I  sec  a;  tan  x  dx  =  sec  x. 

16.  I  cosec  X  cot  .t  da;  =  —  cosec  x. 

Jx     1 
sin-  a;  da;  =  ^  —  -  sin  2  x. 
2     4 

Jx    i 
cos-  a;  dx  =  ^  +  -  sin  2  a;. 
2     4 


INTEGRALS  CONTAINING  Vcr-aj^  (CHAP.  XXV.  AND  ART.  22 
19.      f-J^^sin-^. 


20. 


J'    x-  dx 


•  -Vcr  —  x^  +  —  sin" 


iX'n:(ii;Ai.s  lou  ukm:i;k.\ck  asl 

21.  r   ^^   =iiog ^^-3,^. 


22       i'—^I^L—  —  —  Vrt^j-£- 


23 


24.  ( *  Va?  -  .r^  c7.c  =  -  V^i^^^'  4-  -  sin" '  •^'  • 

25 .  rj;2  Vo^^^  f?x-  =  J*  (2  x2  -  a')  y/TT^^'  +  -  sin->  -  • 
J  o  8  a 

26.  f ^^^ = -g (Art.  227.) 

-^  (,,2  _  ^^.o^t      aVa'^  -  ar^ 

27.  f  (a^  -  a^)'  dx  =  ~(n  tC-  -  2  a-^)  Vo^^^'  +  —  sin-'  ?• 
»/  o  8  a 

INTEGRALS  COXTAIXING  Vx'-  +  U"  (CHAP.  XXV.  AM)  AU  T.  L'JT) 

28.  f      '^'^       =  log  (.X  +  V^^+7?). 


^^^^  =  ^V^?+^- Jlog(.r+ V^?TT»). 
•^  Var'  +  a^     - 

f— ^^=  =  llog ^- =^log 

31.     f       ^_, 


30.      •         ^^ 


a*x 


IxNTEGRAL   CALCULUS     ^^S)      -    *^ 


32 


34.  r  i.^. 


35.     r__*L_  =^__i__ 


36. 


/(.-'  +  a=;i..  =  |(2.^  +  5,.)V?+V+3i*log(.+  . 


ar+( 


INTECxRALS  CONTAn\LVG  V^^'^^^ 


(CHAP.  XXV.  AXD  ART.  227 


37. 


38. 


r     x-dx     _x   ^^ 5     a- 


39.  /-^^^l3,^_,... 

^  XV x'  ~  a-     cc  a 


40.     r__^^L___^vV^-_a_2 


41.     f ^^__  __  Vo;^' -  a-'        1  ^ 


IXrKGKAL.S    FOR    KKIKKEXCK  ;Jba 

42.   JV^r^Vulx  =  I  Var^^^  -  f  log  (.r  +  V^^^^O- 


43. 


44. 


/.I'- Vx'-  —  a-  dx  =  -  (2  a?^  —  ct^  Va;-  —  tr' log  (.c  -+-  Vx*-  —  a'j . 
8                                      8 


*^     to-  /(-\2 


(a-- —  <:r')2  u-'Var^— a- 

45 .     f  (.^-  -  a-)  ^  (Z.v  =  f  (2  a;-  -  o  cr)  V^^^^'  +  ^  log  (x  +  Vr~^^     \ 
»/  8  8 


i^ 


INTEGRALS   CONTAINING  V'2ax-x^ 
fix 


46.     f  =  vers 


47.     I  —  =  —  V 2  a.«  —  x"  4-  ci  vers  '-• 

->'  ■V2ax-ar  " 

c?a;  V2  ax'  —  xr 


48 


cv  V2  aa;  —  ar  ^-^ 

r,  ,  a^       ^_,a; 


49.  f  V2  ao;  -  x'  dx  =  ^^— ^  V2  ax  -  ar  +  ^  vers"'  -  - 
J  2  2  a 

50.  J.W2^^^r:^d.  =  -^^^^'  +  ^--^V2^^^  +  f- 


51.      r-^^^-^i:!^=V2ltarr^'  +  avers-'?. 


884 


INTEGRAL   CALCULUS 
52.      rV2^^^^^__      (2ax~x^)i 


4 


53. 


54. 


dx 


3  ax^ 


J  {2ax~x')i~^,^^Jfa^^: 


IJ^TEGRALS   CONTAINING   ±  ax^ 


+  &.!•  +  C 


55. 


56. 


^U^^^. 


J      2a^+5_V6^^=T 


57 


«c       2ax  +  h  +  -y/urz:^^^- 

/dx  I 


58.  J  V^xqr^^Tp7rf^._2_af^+_&^ 

4  a 


aa;-  +  6a;  + 


-^^  log  (2  aa;  +  6  +  2VaV^^^Tb^:f7y 


4a 


•     8  a^  vW+f^i 


INTEGRALS   FOR   RKi-'EliKNCE  3»5 

OTHER   INTEGRALS 


61.     fJ'^+^dx 


62. 


=  V(a  +  x){b  +  a;)  +  (a  —  6)  log(Va  +x  +  -Vb  +  x). 

fyjj-^^'^^  =  V(a-a-)(6  +  .r)  +  (a  +  ?.)  sin-'  J'"  +  (' • 
J    ^  0  +  X  'a +  6 


INDEX 


Acceleration -[o 

Angles,  between  two  curves    .     .  174  1^3 
■ntth  coordinate  planes    .     .  '  133 

Arc,  derivative  of jyg^  jgy 

Area,  any  surface '355 

derivative  of '    241 

of  curve   .     .   242,  306,  320,'  325,  349 
surface  of  revolution      .      337  353 

Asymptotes '  jyy 

Attraction  at  a  point '    375 


Cauchy's  test  for  convergence 
Centre,  of  curvature 

of  gravity        .     .     . 

of  pressure      .     . 
Change  of  variable 


82 
■     .     .     .     200 

•  .      .      .     oil') 

•  •     .     .     373 
57,  58,  148,  2(j3, 

299,  304,  317 

•  •       li)5,  200 
209 


Curves,  direction  of     .     .     .     ifi^  174^  '^J^o 
I  for  reference,  higher  plane      '  Ki" 

length  of 327,330 

osculating '  20^ 

Definite  integrals 097 

as  a  sum  .  .  .  319 
definition  of  .  .  310 
double  ....  343 
sign  of      .     .     .  324 

Differential  coefifieient      .     .     .     .  jo 

Differentials,  detinitiou  of  .  '  "  eg  70 
formulae 7^ 

Differentiation,  algebraic  formuljB    20,  29 


Circle,  of  curvature     . 

osculating 

Comparison  test  for  convergence .    .      80 
Computation,  hj  logarithms     .     .     .      94 

of  77 qg 

Constant,  definition  of    ...     .  "j 

derivative  of    .     .     .     .'    ,'      26 

.        1 

224,  315 

207,  209 


notatuju  of 
of  integration  . 
Contact,  order  of     ....   20( 
Convergence,   absolute    and  "c( 
tional      .     . 
interval  of 
tests  for     . 
Curvature,  centre  of 

circle  of     ...     ' 
direction  of    .     .     . 
radius  of 
uniform 


definition  of 

inverse  trigonomet- 
ric formuhe      .     . 

logarithmic  and  ex- 
ponential    formu- 


26 


■  ■     .       86 

■  .     .       79 
.     .     .     200 

•  195,  200 

•  .     .     189 
193,  196,  197 

...     193 

variable     ....  P)4 

Curve.'*,  antcle  of  intersection  of"    .'  J74,  1x3 

■area  of  .     .     242,  .306,  .320.  325*  ,34<) 

••outiimous    and    discontinu"- " 

ous  ....  '  00 


order  of       ....     137 

partial i;;o 

successive    ...  61 

trigonometric  formu- 

T^    .       .  ise 45 

Derivative,  definition       ^ 

general  expression  of  .  .  12 
illustrations  of  ...  .  y.i 
meanings  of  .  .  .  lo,  17,  18 
of  an  arc  .  . 
of  area  .  .  . 
of    function    of 

tion 

partial 

partial,  of   higher  order 

136,  137 
relation     between     ^' 

and  iLl  . 


■    16,  17 
.     ■  186,  187 
241,  .3.36 
func- 

.    .     .ns 

.     .     1.30 


57 


38G 


total 


140 


INDEX 


Element,  of  area      .     . 
of  an  integral 
Envelopes,  definition  of 
equation  of 
of  normals 
Equation,  of  envelopes 
of  evolute    .  « 
of  normal 
.  of  tangent    . 
parametric  . 

Evolute       

an  envelope  . 
equation  of  . 
properties  of  . 


20i; 


Function,  algebraic 

continuous 22 

definition 

discontinuous 

expansion  of 

implicit,  differentiation  of 

increasing  and  decreasing 
inverse  trigonometric    .     . 

logarithmic 

of  two  or  more  variables  . 
transcendental  .  .  .  . 
trijronometric 


320 
319 
215 
215 
217 
215 
201 
133 
133 
324 
201 
217 
201 
,204 


Gravity,  centre  of 


Higher  plane  curves Iii2 

Huyghens's  approximate   length  of 
arc O.-. 


Integratiou,  containing  (ax+6)»  .    .  2ii3 
containing 

p  r 

{ii3:+b)9,(ux  +  by       .  2<M 
containing 

V±  x2  +  ax  +  b  ...  266 

definite .Vfl 

definition  of 22^1 

double      .     ;t43,  347.  349,  XM 

evaluation  of      ....  :at7 

for  reference       ....  37s 

fundamental 22."> 

indefinite 309 

of  sec"  X  dx,  cosec"  x  dx   .  274 

of  sin"  X  dx,  cos"  x  dx       .  270 
of  sin™  X  cos"  X  dx 

271,276,291,29.3 

of  tan"  X  dx,  cot"  x  dx      .  273 
of  tan"  X  sec"  r  (/a;, 

cot"  X  cot"  X  dx  274,  291 ,  ilCJ 
of  ti"s\unxdx, 

of  x"»(rt  +  6x")p</j;  .     .    .    2»4 

oiAx^)xdx 2«»5 

of  rational  fractions    .     .    249 
proofs  of  formulae  .     .     .    227 

successive       *13 

triple     ....   "44.  .■«>!,  :«» 

Intercepts  of  tangent 173 

Involute -f>l 

'     properties  of  ....      202,  2<U 


Iiici-cment 11 

Indefinite  integral 309 

Inileterminant  forms 100 

Inertia,  moment  of  ....     .       .34(),  303 

Inlinite,  limits 315 

variables 315 

Infinitesimals,  order  of 73 

Inflexion litO 

Integration  and  derivative  of  integral    270 

^  as  a  summation  ....    •>"•> 

between  limits    ....    '-^M 

by  algebraic  substitution    299 

by  parts 279 

by  substitution  .     .       21S-22<i 
constant  of     .     .     .       224,  :!l(l 
containing 
■y/a;i—x2,   V  ^±  "■■^  •     -•"' 


93.  ;?27, 


Leibnitz's  theorem  .  . 
Length  of  curves  .  . 
Limit,  change  of 

definition  of 

infinite     

Napierian  b:isc 

notation  of 

relation  of  arc  to  chord      .     . 

variable :M3.  .US. 

Liquids,  pressure  of 

I^jgarithmic  functions     ....       2 
Logarithms,  computation  of     .     .     . 
Napierian 


Maclaurin's  fhcon-m 
Maxima  and  minima 
.Moment  of  inertia    . 


114.  l.V. 

346,  ;ni;i 


388 


INDEX 


PAOE 

Napierian  base  .       . 10 

Nurmals ....     133 

Order,,  of  contact     .......     206 

of  differentiation 137 

of  integration 343 

Osculating  curves -08 

order  of  contact  of       .     .     209 
Osculating  circle,  coordinates  of  centre 

200 
radius  of 209 

Pappus,  theorems  of 369 

Parameter        214,  324 

Parametric  equations        324 

Power  series 85 

Pressure,  centre  of 373 

of  liquids 370 

Rates 18 

Reduction  formute 284,  291 

Remainder,  Taylor's  theorem  ...     105 

Series,  computation  by 94 

convergence  of  power  ...  85 
convergent  and  divergent .  .  78 
of  positive  and  negative 

terms 78 

power       85 

Slope  of  a  curve 16 

of  a  line 16 

of  a  plane 133 

Subtangent 173,  183 

Subnormar 173,  183 


PAr 

.     355 

337,  353 


Surfaces,  area  of  any      .     .     . 
Surfaces  of  revolution,  areas  of 

derivative  of 
area  of  .     , 

volumes  of    , 


Tangent 70 

intercept  of 173 

Tangent  planes J"<o 

Taylor's  theorem     ....     97,  103,  145 

Theorem,  Leibnitz's 65 

Maclaurin's 89 

mean  value      ....     84,  ^6 

Pappus's 3()9 

RoUe's S3 

Taylor's        ...     97,  103,  145 

Transformation       152,  153 

Trigonometric  functions       ...       2,  45 

Uniform  curvature       ......     193 

Unit  of  force       .     ..o     ....     376 

Variable,  change  of     .     .     -     57,  58,  148. 

263,  299,304,  317 

curvature     .....    194 

definition  of 1 

dependent    2 

independent      2 

notation  of       1 

Velocity 17,  18 

Volumes,  any  solid 361 

by  area  of  section    .     .     .    340 
surfaces  of  revolution     333, 353 


c>^  %. 


^2fyf^  A  *^  ^Sv-rr-rt^ 


.y^  /^  x/  r^ 


